Love is a very common theme in Shakespeare's comedies. The structure of a classical Comedy is such that, after a series of complications for at least one romantic couple, the play ends "happily" in at least one marriage. Though not everyone actually says his/her vows before the play's end, Twelfth Night concludes in three marriages -- Viola and Orsino, Olivia and Sebastian, and Sir Toby and Maria.
Love is depicted in a number of ways. First, there is the melancholy, unrequited "sickness" that is Orsino's love. He opens the play mooning over Olivia and the fact that she will not return his love. He seems to be the sort of lover that actually takes a sort of delight in his own misery. Viola mirrors Orsino when she dresses as Cesario and takes on the role of a young man serving in Orsino's household. She falls hopelessly in love with Orsino and describes herself this way:
...[S]he never told her love,
But let concealment like a worm i' th' bud
Feed on her damask cheek: she pin'd in thought
And with a green and yellow melancholy
Sat like Patience on a monument,
Smiling at grief.
Olivia, unable to return Orsino's affections, falls in love with Cesario (Viola), and her love is a sort of unrequited sickness as well. However, when Olivia mistakes Sebastian for Cesario, the two are wed, really even before Sebastian knows what has happened to him. Sebastian's love is "love at first sight" in the extreme.
Then there is Malvolio, who fancies himself in love with Olivia, but seems, rather, to be very much in love with becoming the lord and master of her household. Sir Andrew also falls into the "unrequitedly in love with Olivia" category. Shakespeare uses him to spoof the actions of a young courtly man -- pursuing her through making good friends with her male relative (Sir Toby) and challenging his rival (Cesario) to a duel.
And, though Sir Toby and Maria (Olivia's serving woman) never have any scene in which they confess their love for each other, it is reported at the end of the play that Toby was so grateful to Maria for the part she played in duping Malvolio that "'[i]n recompense. . .he hath married her." Is there love between them? Shakespeare doesn't answer this question.
The links below will connect you to essays that further investigate the topic of love in Twelfth Night.