Hans Christian Andersen was a Danish author. Andersen is of course, best remembered for his fairy tales, although he was also a prolific writer of plays, travelogues, novels and poems. And where many would assume his tales are simple children's stories, they hold much popularity amongst many adults. 

In total Andersen published 3308 tales which have been translated into more than 125 languages. Some would say that  they have become culturally embedded in the West's collective consciousness as they are so readily accessible to children, but also through presenting lessons of virtue and resilience in the face of adversity for mature readers as well. Some of his most famous fairy tales include "The Emperor's New Clothes", "The Little Mermaid", "The Nightingale", "The Snow Queen", "The Ugly Duckling", "Thumbelina", and many others. His stories have inspired ballets, plays, and various movies and TV series.
Rapunzel

The character of Rapunzel comes from the Fairy-tale of the exact same name, found in the Brothers Grimm collection. This coming of age tale tells of a long-haired girl locked in a tower by a cruel witch who kidnapped her when she was still a baby. The story complicates however, as the two do have a mother-daughter relationship that is love/hate to say the least. As Rapunzel grows to become a young woman she desperately wishes to see the world, and interpretations of the tale vary from Rapunzel escaping with a prince who discovers her tall prison, to the Witch discovering her romance with the prince and blinding him with scissors. In Lapine's interpretation of Rapunzel's character, her sad story serves as a message about the child abuse many suffer. It also carries the theme of growing up, and teenagerhood. According to scriptwriter Lapine, "A lot of fairy tales are about maturation, and Rapunzel represents the teenage girl who wants to sneak out of the house and have a life of her own, out from under her parent's thumb. Many times teenagers who have already gone through puberty feel bound; they feel adult, even though they're not, and want to do adult things, so this storyline is still very relevant today."

Cinderella
Cinderella, a tale that is also known as Ashenputtel or The Little Glass Slipper, is a folk tale embodying a myth-element of unjust oppression and triumphant reward. Cinderella is one of the older and most popular existing folk tales today and has had many versions over the years. Lapine's interpretation of Cinderella is interesting because she is not painted as as much of a victim as many versions do. Of course in the Disney tale, there is no sign of her father, she is abused by her step mother and step sisters but then because she's such sweet, unassuming young lady a fairy godmother appears and gives her everything she ever wanted. However, Into The Woods challenges this, from the very start she seems a lot more real and human than at all princess like. Immediately appearing scrubbing the floor, looking indeed a pitiful site, she sings that she wishes "more than anything" and more than life itself to go to the king's festival taking place at the palace. Put in this context, suddenly the whole Cinderella story seems a bit backwards. Why after being treated so poorly all your life would you wish to go to a party? Surely, the desire for basic human needs such as love and touch and affection would be much greater. Thus, Cinderella is revealed to be a frivolous, indecisive (see: the entirety of Steps of the Palace) and occasionally spineless, although at heart still a sweet and kind young woman.

The Baker’s Wife & The Baker

Though the Baker and the Baker’s wife were created by Sondheim and Lampine, the characters may have been inspired by Hans Christian Andersen's fairy-tale, Thumbelina. It is in this tale that a childless couple seek the help of a woman with magic powers, asking her for help with their infertility. The woman gives the couple a barleycorn to plant, and from the flower of that plant emerges little Thumbelina. This of course heavily inspired Sondheim and Lapine's tale of the childless baker and his wife, as the spell that enables them to have their child requires hair as yellow as corn, or, the hair OF the corn.

Little Red Riding Hood and The Wolf
This story’s origins can be traced back to the 10th century by several European folk tales, including one from Italy called The False Grandmother, and from France called Little Red Cap. The best known versions are those written by Charles Perrault and the Brothers Grimm.

For many years this tale has served as a fable, warning (and scaring) children of the dangers of disobeying your mother. Some interpret Little Red Riding Hood’s story as a sexual coming-of-age story. In Into the Woods, the Baker and his Wife seek the iconic blood-coloured cape from Red Riding Hood to give to the Witch to break her evil spell on them. James Lapine has stated, "Little Red Riding Hood is a naive, pubescent little girl wearing a red cape, which I believe represents her passage into adulthood. Until the Wolf comes along, she believes that everything and everyone is good in her life and everyone is to be trusted”

Although varied from tale to tale, the wolf broadly represents danger. Some, darker or more sexual than others. However in all of the tales the wolf comes to a different vile end. He usually killed by either a hunter or a woodcutter and has been made into a cape, a stew, and has even been cut open, filled with stones and then sewed up again whilst still alive. This Lapine references in the script, just before the song I Know Things Now takes place.
Jack and the Beanstalk 
is an English fairy tale. In 1734 it was published as "The Story of Jack Spriggins and the Enchanted Bean" and as Benjamin Tabart's moralised "The History of Jack and the Bean-Stalk" in 1807. Henry Cole, publishing under pen name Felix Summerly, popularised the tale in The Home Treasury(1845), and Joseph Jacobs rewrote it in English Fairy Tales (1890). The most commonly reprinted version is Jacobs' simply because it is believed to be closer to the oral versions than Tabart's because it lacks the moralising. "Jack and the Beanstalk" is the best known of the "Jack tales", a series of stories featuring the archetypal Cornish and English hero and stock character Jack. According to researchers at the universities in Durham and Lisbon, the story originated more than 5,000 years ago, based on a widespread archaic story form which is now classified by folklorists as ATU 328 The Boy Who Stole Ogre's Treasure. In into the woods Jack is a young, dim witted boy on the cusp of puberty (around age 12), just like Little Red. The brave element to his character is clearly influenced by the original tales.
More inspiration for Into The Woods lies in psychology; Sondheim was a fan of the works of the Swiss psychiatrist Carl Jung...

“I wish my son were not a fool.
I wish my house was not a mess.
I wish the cow was full of milk.
I wish the house was full of gold-
I wish a lot of things...”

The baker’s wife then proceeds to let in Little Red Riding Hood who-in her naivety, asks both of them for some bread to give to her sick Granny before she carelessly wanders into the woods. We can see from the way it is sung that she is excitable and jumpy, this shows she has the general optimistic curiosity you would expect from a girl of a young age in her situation. She then keeps asking for more, clearly not understanding social boundaries and proceeds to insist for “A sticky bun? Or four?” which many perceive as somewhat brattish behaviour, and adding to the image that she is quite an irritating character. It then appears that Cinderella can apparently talk to birds and get them to do her extra chore so she might go to the festival and indeed the ball. Jack then sings, encouraging his cow Milky White to produce milk so that his mother doesn’t sell her. This helps to show Jack’s character as optimistic, hopeful and caring – if unintelligent. Unlike his mother who views Milky White as solely an asset to be used for food and money, he is shackled by emotion and his attachment to his friend regardless of her apparent ‘use’.

After asking further for even more food with her mouth full of what she needed for her “Poor old Granny”, Little Red Riding Hood begins her journey into the woods. When she utters the following it is a clear indication of what her obstacle is; her naivety.

The way is clear,
The light is good,
I have no fear,
Nor no one should.
The woods are just trees,
The trees are just wood.

Such an extreme fearlessness ironically almost tells the audience straight away that she is in fact vulnerable and trusts people way too easily, before asking the Baker and his Wife (who are still strangers to her) for a basket as well, adding insult to injury.
The song continues and The Witch appears to the Baker and his Wife, coincidently being their neighbour meaning they are familiar with her. Terrified, they immediately blurt out that they have run out of bread. The Witch of course, cares not for their bread and instead 

In this song, Sondheim takes Little Red Ridinghood on a journey from innocence to sexual awakening. This theme is prevalent in the musical, with Jack going on the same journey of growing up with strong themes of sexuality in the song Giants In The Sky. The story of this song is as follows:
Stanza 1 -
Little Red sings of disobeying her mother's instructions by straying from the path. She knows that this was wrong, but explains that The Wolf (who persuaded her to stray) seemed rather nice.  The way Sondheim has used four short rhyming phrases in the first two lines creates a simple, repetitive and lively rhythm immediately at the very start of the song. This instantly reflects the simplistic nature of Little Red's thought process as she is a young girl. However, this stanza also demonstrates her (albeit new-found) maturity, as she is self aware enough to know that she has done wrong and should have done better, also knowing that her mistake was trusting The Wolf simply because he appeared "nice".
Stanza 2 -
In this second verse the song begins to pick up a flowing melody that rises up and down in pitch quite merrily. It is also at this point in the song that it becomes just as much a stream of consciousness as it is her telling a story, because whilst she sings she relives the events. This is shown through the repeated use of the word "and"