Sebastian is Viola's twin brother who has been lost at sea. Off-stage, he's separated from Viola at but survives the ship wreck by clinging to the ship's mast. Eventually, he's fished out of the ocean by a sailor, Antonio, who falls in love with him.
We don't see much of Sebastian in the play, but his character is significant in Twelfth Night. Once Sebastian travels to Illyria, he's mistaken by all for "Cesario" and quickly hooks up with the Countess. Sebastian's unlikely marriage to Olivia allows her to redirect her desire for "Cesario" into a sanctioned heterosexual relationship. As we've seen before, Twelfth Night is a comedy and, as such, it works its way toward marriage and the reunification of families.
That said, Sebastian clearly has a close relationship with Antonio, who may or may not be a lover. In this way, Sebastian's relationship allows the play to study, briefly, the erotics of male bonds. (You can go to "Quotes" for the theme of "Love" if you want to read about this in more detail.) Antonio's relationship with Sebastian recalls that of Antonio and Bassanio in The Merchant of Venice, where homoerotics and male friendship are explored in much more depth and detail.