What is the importance of Antonio in Twelfth Night?
In his comedies, not only does Shakespeare explore the trials and tribulations of romantic love, he also explores the nature and constancy of masculine bonds of love or friendship. In Shakespeare's day, male friendship was very important throughout a man's life -- his male friends would be his companions in all the events and work of his life outside the home, often considered more important than the bonds of romantic love.
So, in Twelfth Night we have the friendship between Antonio and Sebastian (just as in The Merchant of Venice we have Antonio and Bassanio and in The Two Gentlemen of Verona we have Proteus and Valentine). There is a bond of loyalty between Sebastian and Antonio, with Antonio entrusting his purse of money to Sebastian. He displays all the openhearted generosity of a perfect friend to Sebastian.
It is, however, the comic mistaken identities that cause Antonio to think that his friend has betrayed him by not returning his purse when it is required. So, not only does he establish the theme of male friendship, he is key to maintaining and continuing the plot of mistaken identity.
Of course, this is a Comedy, so at the end, all is rendered right, as Antonio finally sees Sebastian and Viola together and understands that it was not Sebastian's lack of loyalty and friendship, but merely a mistake of identity. All is set right, as it should be, at the end of the play.