Theseus is the Duke of Athens, and consequently the most powerful character in the courtly realm of the play. Though he's missing entirely from Acts 2 and 3, his upcoming wedding to Hippolyta is the subject of the play's opening and closing acts.
Theseus also has the privilege of the play's opening line, which is always a big deal in Shakespearean drama. In this case, the opening speech sets forth the play's biggest theme, love:
Now, fair Hippolyta, our nuptial hour
Draws on apace. Four happy days bring in
Another moon. But, O, methinks, how slow
This old moon wanes! she lingers my desires (1.1.1-4)
Translation: Theseus can hardly wait to spend the night with his bride-to-be—"four happy days" seems like an eternity when you're in love. To make the time pass quickly, Theseus tells his party planner (Philostrate) to go and "stir up the Athenian youth to merriments" (read: partying, hooking up, and fooling around, which is basically what goes down in the woods).
Theseus is more than a guy who likes to celebrate. As the Duke of Athens, he's also the play's resident Judge Judy, so he's all about upholding law and order. This is why Egeus turns to him when his unruly daughter (Hermia) refuses to marry the guy he's chosen for her. In this case, Egeus begs Theseus to uphold Athenian law, which dictates the death penalty for disobedient children:
I beg the ancient privilege of Athens,
As she is mine, I may dispose of her,
Which shall be either to this gentleman
Or to her death, according to our law
Immediately provided in that case. (1.1.42-46)
Theseus's response is the subject of some debate. As we know, he offers Hermia two choices if she refuses to marry: the death penalty or life as a celibate nun. Some audiences view this as evidence of Theseus's benevolence and compassion, because he offers Hermia a choice other than marriage or death. The argument goes something like this: In offering Hermia an alternative and encouraging her to think long and hard about her choice, Theseus is being sensitive to her feelings, and though he won't bend the law for her, we get the sense that he truly wants Hermia to be happy. If you like this argument, then you'll also want to think about the fact that, eventually, Theseus overrides Egeus's wishes and grants Hermia and Lysander permission to marry.
On the other hand, some audiences and literary critics think this line of reasoning is a crock of bull. Such critics argue that Theseus doesn't actually offer Hermia any positive choices in the matter. Further, he tells her that being a lifelong virgin is pretty much the same as a death sentence because celibacy is like "withering on [a] virgin thorn" (1.1.79). The marriage/death/nun thing also seems like a blatant attempt to control Hermia's sexuality. If she won't marry the guy of her father's choosing, then she'll die or she won't have sex with anyone. Ever.
The circumstances surrounding Theseus's marriage to Hippolyta are also an important subject of discussion. We learn early on that Theseus and Hippolyta are engaged because Theseus conquered the Amazons and basically helped himself to their queen. (That would be Hippolyta.) On the one hand, this whole affair suggests that lovers and spouses can be taken by force, which is what Hermia is trying to avoid. This idea resurfaces throughout the play, especially when Titania informs Bottom that he's going to be spending a lot of time with her, regardless of whether he wants to or not. At the same time, we can point out that Hippolyta seems pretty happy being engaged to Theseus. She even promises him that he's in for some fun on their wedding night (1.1). Read our analysis of her character for more on this.