How are different types of love explored in "Twelfth Night"?
Firstly, we certainly see a great deal of "romantic love" in Twelfth Night. The play explores unfulfilled romantic desire in the play's love triangle between Orsino, Olivia, and Viola/Cesario. At the end of the play, we see the fulfillment of romantic love with the coupling of Orsino and Viola and Olivia and Sebastian. The most true romance we see between two characters, however, might be in the flirtatious and seemingly truly mutually affectionate relationship between Maria and Sir Toby that does, ultimately, end in a report of marriage.
The play also explores sibling or familial love. We see the grief caused by the loss of a sibling for both Viola and Sebastian—who believe the other drowned—and for Olivia, who is mourning the death of her brother at the play's beginning. The ties between family are further explored in the relationship between Olivia and Sir Toby—it is a tense relationship as Toby's drunkenness and brawling are problematic in Olivia's household. Olivia responds to her uncle's behavior with a mixture of care and concern, anger, and sadness over the course of the play.
Friendship is a type of love that is complicated within Twelfth Night. There is a sort of friendship between Cesario and Orsino that is complicated by Viola's feelings for her boss and Orsino's confusion over his feelings for Cesario. There is also a sort of friendship between Feste and Olivia and Maria and Olivia, but this is complicated by status. The friendship between Toby and Andrew is complicated by the fact that Toby uses Andrew for his own amusement. The only true friendship to be found in the play is that between Antonio and Sebastian—and there is scholarly contention that Antonio's feelings for Sebastian are more romantically inclined than anything else.
Finally, this play deals with the idea of self love. This is most evident in the character of Malvolio, whose love for himself is his own downfall. His opinion of himself is so elevated that he is blind to the trickery of Toby and Maria and ends up disgraced and humiliated.