Aoimirai - Sen to Chihiro no Kamikakushi (Spirited Away) explained

Sen to Chihiro no Kamikakushi
This article tries to explain most of the analogies, symbolism's and folk tales that are everywhere in “Sen to Chihiro no Kamikakushi”, which actually translates into “The disappearance of Sen and Chihiro”, where “disappearance” is “spirited away”. Being spirited away is on itself a tale of children who vanish for a few days only to be found perfectly fine later but with gaps in their memories, because they were “spirited away” into the spirit world temporarily. 
For ease of reading, we composed this in a bullet point article that follows the timeline of the movie. Please notice that pretty much every official translation (either dub or sub) get a lot of things wrong, so if you want to see a better subtitling, it is highly recommended to watch from a Fansub version.

1. The small shrines in the road they take “as a shortcut” are little Shinto Shrines for nature and supernatural spirits. It hints that the road ahead is one of intense spiritual power.

2. Chihiro, still saddened by losing her old friends and school (and from the 5 stages of grief, leaving denial into anger as she is bothered that her first bouquet is one of goodbye), is careful about her surroundings: Children are more perceptive to the supernatural than adults.
3. While her parents perceive the large red entrance as the old entrance to a now-abandoned theme park, Chihiro feels otherwise, specially by the moss covered spiritual symbols, which show no human presence been there for a long time. The door also inhale the air despite the bland weather, showing a flow from other realm. After getting past it, Chihiro mentions the wind "sounds like a cry".
4. Chihiro is scared by the multiple hints her intuition give about the place, while her parents are embroiled in the wonder and smells of their senses
5. The river they cross to get to the stores represent the division between real world and spiritual world. Rivers are standard divisions on most oriental folk tales.
6. Her parents, "with knowledge", believe they are ok starting to eat on the banquet around them, as they have "cards and money", while Chihiro denies, intuitively realizing this is not a place where their card or money have actual value. Also, note how her parents are extra gluttons now (an effect of non-pure spirits in this realm. Chihiro, as a child, is unaffected)
7. Walking around, Chihiro stumbles upon the Bathhouse, the center of the local spiritual presence. A humanoid spirit - Haku - sees her and immediately realizes she is a human: Humans cannot exist in the spiritual world and would be "eaten" by the spirits for their strength and power. He urges her to flee. Note: there is speculation that the Bathhouse also represents a brothel, but this is only because of the appearance of the building with a real historical brothel, and Yubaba nearly dictatorial work schedule, whose most attendants are female-looking spirits (which, in fact, have no real gender, the “female-like” are snakes, and the “male-like” are frogs. Again, a common spiritual representation of genre). The meaning of work and self worth, while a critique to capitalism and current society, remains the same.
8. As the night sets, Chihiro is now entirely in the spiritual world. In it, her parents are turned into pigs, representing their gluttony. They are now completely irrational and driven by their instincts. Chihiro, in despair, tries to leave, only to find that the river has now raised and blocked her way out of the spiritual world.
9. The river, the border of spiritual and human world, is overflowing with life force. Spirits from all parts come to visit the Bathhouse and stores by the river. One can see multiple deities or representations in this sequence.
10. Despair sets in and Chihiro cries. Humans cannot exist in this realm, and she is disappearing. Haku finds her, and she can no longer touch him. Realizing she is disappearing, he gives her something from the spiritual world to eat. That makes her now part of the spiritual world, and she recovers her presence of spirit (literally). 
11. Fear still binds her to the ground, and they don't have time to lose, so Haku uses magic to unbind her from the grass, giving her strength.
12. Haku tries to get Chihiro to the safety of the Bathhouse by making a spell that renders her human presence unknown. Breathing (symbolism for life) would reveal her despite the spell.
13. NoFace appears for the first time. NoFace is just like his name implies. He is a pure spirit with no will or desires, and such, absorbs anything. However, he takes notice of Chihiro for her humanity, purity and will, something different from what he is used in that realm. He is also forbidden of getting into the Bathhouse (as he would absorb the strong energy ~mostly negative~ and become uncontrollable ... as we eventually see happening)
14. Haku explains that to live and eventually save her parents, Chihiro must remain hidden from the spirits, or work for the Bathhouse owner (and thus be protected under her contract), Yubaba, an entity with many meanings, but mostly an insidious witch, owner of everything around and usurper of power and money. A basic capitalist figure, she cannot refuse those who ask for work, so, Haku argues, she will trick Chihiro in trying to leave rather than work (if she doesn't work, Yubaba will rather just turn her into pig or whatever)
15. Haku knows Chihiro's name from the "past", which also explains why he is so keen to help her ... what past is only revealed later.
16. Sootballs work for Kamaji, because everyone in this world must work (or be useful), even soot.  (Sootballs are based in those shown in Tonari no Totoro)
17. Kamaji convinces Lin to take Chihiro to Yubaba, as Kamaji can't have her working there. Lin is scared because the presence of a human is already know and she doesn't want to be caught protecting her.
18. The big spirit who eventually helps Lin and Chihiro in the elevator is clearly a Radish Deity Spirit.
19. As expected, Yubaba tries to dissuade Chihiro from working, and tries to scare her into acceptance.
20. "Baby" represents the huge importance of a child, making even the most powerful of witches answer for his whim. Busy with "Baby", Yubaba have no option but to hurry and rush a contract.
21. By signing the contract, Yubaba "steals" her name, Chihiro is now owned by Yubaba, represented by loosing her name. For a new name, Yubaba steals 3 kanji from her name, leaving only the one for Sen. Chihiro is bound to Yubaba until she allows, or Sen can remember her name (representing she is set free).
22. Chihiro thinks there are two Haku because he treats her coldly, not recognizing he is just pretending (Yubaba would be furious if he knew her)
23. Yubaba flies in the dawn to the horizon. It hints us that despite her differences, she still visits family (Zeniba) as if in neutral ground, though she uses this time probably to also spy on her.
24. Having helped the Sootball the previous night, showing kindness to those who are nothing but "dust", they are immediately friendly towards her and keep her shoes (the night before, also protect her from the nasty works of Kamaji). Her sincerity and kindness will be the main drive into making others help her in this world filled with greed. Again, an analogy to our own world.
25. NoFace is still at the bridge, as he has no deal with the Bathhouse (alas, with no one). He keeps getting more interested in Chihiro, which is as foreign as he is.
26. When talking to her parents (pigs), Chihiro is already in full submission as Sen, but Haku gives her clothes back and she reads her name from a card, remembering it. Finally, Haku explains the secret behind "stealing the names", which represent one’s identity.
27. After a long time fighting in her situation, once Haku offers her kindness from a pan, she breaks up in tears for the first time: It is ok to cry, you need to let it out in order to keep strong.
28. Chihiro lets NoFace in unaware she shouldn't. Without his own will or thoughts, he just gets in and further develops an attachment for Chihiro, but also starts absorbing the ill energies from the place. 
29. Yubaba realizes the rain might not be a coincidence, and related to some strong guest (which we eventually learn to be true, it is a river God - thus the rain)
30. Being attracted to Chihiro, NoFace helps her when she needs tags for the herbal bath (to clean the main tub). He further keeps stalking her, unaware that his "empty shell" of sorts is absorbing the ill energy of greedy spirits of the bathhouse.
31. Yubaba also realizes they have an intruder (NoFace), but can’t pinpoint it.
32. While a bathhouse, they still try to stop the huge stinky spirit to avoid losses, after all, Yubaba is all about greed and power. As he approaches, she realizes there is some power underneath all the filth and decides to allow him to approach.
33. Still trying to get Chihiro to quit, she orders her to handle the stinky spirit (before, it is mentioned that from “orders from above” Chihiro is to clean the main (filthiest) tub).
34. Black transparent goo usually represent bad vibrations or simply negative energy. You can see how the stinky spirit is covered with mud and bad vibrations which can even corrupt Lin's breakfast.
35. After entering the bath, the stinky spirit looks at himself realizing he is still not clean, so he asks for Chihiro. Meanwhile, Yubaba is enjoying seeing Chihiro suffer.
36. The stinky spirit takes Chihiro closer, and she realizes he has further dirt that not even the hot bath can clear. As Yubaba hears Chihiro mentioning a "thorn", she realizes the stinky spirit is filled with dirt and, thus, is probably a powerful entity (only a powerful entity can get equally powerfully stinky), so she decides to help Chihiro and enlists the help of everyone in the bathhouse.
37. As everyone pulls the bicycle out, the whole filthiness is released, revealing the River God’s true form. Needless to say, this represents how humans pollute nature.
38. The river God awards Chihiro with a dumpling. It represents the most pure form of health, purity and energy (though Chihiro perceives it as not tasty - she doesn't need it!)
39. As the greedy employees try to find more gold. NoFace, with his strong influence, can show people what they want (even though it is fake). The greed ends up getting the best of him and he starts consuming everyone in a reflex of others consuming gold. As he eats others, he gets their personality and abilities, including the ability to talk. NOTE: in this spirit realm, everyone are spirit essences, so "eating" one another equals consuming one’s energy. You can conversely release the energy back into the original creatures.
40. Chihiro dreams of trying to give her parents the dumpling. Deep in her mind, she knows the cleansing power of the dumpling.
41. The bathhouse is in full work to serve the "rich customer" ... NoFace, getting more and more powerful, while still mirroring everyone's greed. Since Yubaba is asleep, only Chihiro is suspicious of the new customer.
42. Paper birds (an usual witch familiar) attack a Dragon, which Chihiro calls Haku - as she can see other's true form (that is actually a direct answer to her question "what if I can't recognize my parents?". She can, for they are NOT pigs)
43. NoFace is curious how Chihiro is the only one not interested in gold, which makes him unstable for she is the reason he is there. As such, he goes berserk. 
44. One paper bird sticks to Chihiro as a spy for whoever sent them (Zeniba). As Chihiro searches Haku, she sees that Yubaba is not asleep, but was away.
45. Curious to the human intention, the paper bird (or rather Zeniba, who controls it) helps her.
46. Yubaba is now aware that NoFace is present and wrecking havoc, yet ...
47. … she put Baby to sleep before. The greedy powerful witch Yubaba is still a slave to her baby.
48. According to Yubaba's sister (Zeniba), Haku stole a seal. Seals are almost as important as names since they are used as signatures in Japan. They believe the seal had strong protective magic, which is killing Haku.
49. Upon holding Haku's horns, Chihiro sees herself in a river. Eventually we will learn why. The connection they share allows Haku to regain consciousness and flee for Kamiji.
50. Chihiro knows the dumpling is powerful and gives part of it to Haku, which works to purify him and expel the spell.
51. Chihiro squashes the spell, which in this realm can take visible form. EVERY form of spiritual energy can take form in the spirit world.
52. Since Chihiro squashed the spell, she is now poisoned by it unless she breaks the spell immediately. The gestures they perform cutting the representation of the spell in half is a common folktale about preventing spells to take root.
53. Chihiro wants to return the seal to Zeniba to prevent her anger over Haku. Her integrity and sincerity brings everyone to help her, even spoiled Boh (Baby)
54. As much power Yubaba have, she can't either dispel NoFace (because he is a pure empty shell), nor recognize that the rat is her baby (because deep down she cares more about gold than Baby, she is a witch after all). 
55. Again, Chihiro uses the dumpling by offering it to NoFace, somehow aware its purifying powers will cleanse him of all greed. It works immediately making NoFace start losing corruption (black goo) and the spirits he ate.
56. Despite all trouble, Chihiro still calls NoFace to make him get away from the bathhouse. As he vomits the last spirit (which was the first he ate), he loses the ability to talk.
57. Although it was shown before with images from the stores, the other passengers in the train are just shadows. The train represents our lives and how every other person are just "passengers" we might not even remember their faces or characteristics, only a shadow in our memory. Same as the customers on the stores (you can notice this on the scene where the stores are closing when the stinky spirit is arriving)
58. The whole train trip is rife with analogies about our life. Passengers coming and going, the vast "sea" that is the real world, random images and lights passing by. Also, notice how Kamiji mentioned this train is 'one-way only'
59. During the trip, we see a strong willed Chihiro. From all the hardships, she grew and learned fast.
60. Haku mentions that Yubaba was played, the first thing that comes to her mind is the gold (not Baby). As she tries to close Baby's mouth because he is eating loudly, her spell breaks Zeniba spell and the three heads return. Only then NoFace lack of power finally catches up and the gold turns back to what it really was: mud.
61. Yubaba calls Zeniba an evil witch. Zeniba calls Yubaba an evil witch. It goes to show that we are usually hostile with those we don't understand. For one, others are evil if they don't agree, but in the end, it is all about different point of views.
62. Zeniba explains to Chihiro that the spell she squashed was actually Yubaba's control over Haku. That explains why he challenged Yubaba earlier telling he will return Baby if she returns Chihiro and her parents to the human world.
63. In the cozy home of Zeniba, NoFace doesn't turn into a greedy creature, but rather a helpful one, showing both how he absorbs the environment energies, and that Zeniba is, indeed, Yubaba's opposite (Even Zeniba tells her sister is her "opposite").
64. Zeniba gives a hair band to Chihiro. This will be extremely important in the end of the movie because it is proof everything was not a dream. Zeniba also mentions the hair band will protect her because it was made from the thread they (friends who care about her) spun: they all care for Chihiro, and therefore, their positive energy is in the hair band. A "spell" can be something as simple as a good memory, that when you remember, gives you energy and hope.
65. Knowing that the Bathhouse environment is negative to NoFace (which absorbs energy), Zeniba invites him to stay as her helper.
66. As they fly back, Chihiro once again sees herself in a river while riding Haku's dragon form (his true form). She finally remembers when she almost drowned in a now extinct river and realizes that river is Haku, she remembers the name of the river and tells Haku, which is now free, since he remembered his name. And this is why they both knew each other.
67. Yubaba dares Chihiro to tell from a group of pigs which are her parents. This is a logic test, a rather easy one at that: Her parents are not pigs, thus they are not there. Also, it would be impossible to differentiate them while in pig form, so the only logic possible answer is that they are not one of them. Also, Chihiro’s strong intuition can discern the pigs, just like she knew the dragon was Haku, or that NoFace is not intrinsically evil.
68. Haku can't return with Chihiro because he doesn’t exist on the human realm anymore, the river (he) no longer exists in the real world.
69. Chihiro is not supposed to look back: it represents regret and longing, and that would undermine her will to really leave.
70. Initially, when they came in the tunnel, Chihiro held her mother for protection. Now, stronger and grown up, Chihiro holds her because she doesn't want to let her go.
71. The car is dirty and full of leaves, representing the passing of days, not hours, as her parents expected. As she turns, we see the hair band she got from Zeniba sparkle.
72. Her Journey into adulthood is over.