Hippolyta is the Amazon Queen who marries Theseus (off-stage in Act 4, Scene 1). Shakespeare bases Hippolyta's character on the ancient historian Plutarch's portrayal of her in his "Life of Theseus," which covers the big, mythological battle between Theseus and the Amazons. (The Amazons are a mythological group of warrior women.)
Before you get too excited about the prospect of Shakespeare developing a fierce female role, we should warn you that Hippolyta is kind of a dud in this play. When A Midsummer Night's Dream opens, she's no longer the warrior woman she used to be because she's already been conquered by Theseus, who happily reminds her "I wooed thee with my sword, / And won thy love doing thee injuries" (1.1.2). As far as we can tell, Shakespeare's Hippolyta doesn't mind being a literal trophy wife – she seems happy enough to be engaged to Theseus and even looks forward to her wedding night, when "the moon [...] shall behold the night / of [their] solemnities" (1.1.9-11).
For a lot of feminist scholars, the implications of this are pretty important. Symbolically, Hippolyta represents female power that's been suppressed by male authority, which is a recurring theme in the play. (Although Hippolyta's defeat has already occurred off-stage, we can see it being reenacted by the power struggle between the Fairy King and Queen, Oberon and Titania, which results in Titania being humiliated and whipped into submission by Oberon.)
Some literary critics see Hippolyta as an allusion to Shakespeare's own ruler, Queen Elizabeth I, who was often imagined as an Amazon warrior in literature and art. Elizabeth was notorious for refusing to marry and give up her power as England's first female monarch. Louis Adrian Montrose suggests that Hippolyta's submission to Theseus enacts a common Elizabethan fantasy about making a powerful female ruler submit to male authority, which is exactly what Elizabeth refused to do when she declared her intention to remain unmarried (source). You can read more about Hippolyta's status as an Amazon by going to "Themes: Gender."
Aside from the implications of Hippolyta's submission to Theseus, this bride-to-be doesn't play a major role in how the plot unfolds. Like Theseus, Hippolyta disappears after Act 1, Scene 1 and doesn't return to the stage again until her wedding day in Act 4, Scene 1. When she does resurface in the play, it seems to be in the service of Shakespeare making allusions to classical mythology. [For example, when she's out hunting with the wedding party, she tells everyone about the time she once went hunting with "Hercules and Cadmus" back in Crete (4.1.1).]
Hippolyta also seems to be a bit of a romantic. (Titania mentions one of her sexual affairs when she accuses Oberon of having a steamy romance with "the bouncing Amazon" (2.1.2). Also, Hippolyta is one of the few people who actually believes the young lovers' account of their zany night in the woods.) Even though Hippolyta is sympathetic to the young lovers (all nobles), she turns out to be quite a snob. She bags on the Mechanicals throughout their bumbling, amateur performance of Pyramus and Thisbe and even says "I love not to see wretchedness o'er charged / And duty in his service perishing" (5.1.3). Translation: Hippolyta hates to look at poor people, especially when they're such lousy actors.