Describe the relationship between Sebastian and Antonio in Twelfth Night.

Antonio looks after Sebastian.  He really cares about Sebastian, and wants to protect him.  This is why he stays in Illyria even though his legal history is such that he really should not be seen around there.
Antonio seems to think that since he rescued Sebastian from the shipwreck he should continue to look after him.

SEBASTIAN
O good Antonio, forgive me your trouble.
ANTONIO
If you will not murder me for my love, let me beyour servant.
SEBASTIAN
If you will not undo what you have done, that is,kill him whom you have recovered, desire it not. … (Act 2, Scene 1)

Antonio gives Sebastian his purse to hold.  They agree to meet up later.  It is all fairly innocent except there is the one problem.  Antonio is a wanted man in Illyria. 
Sebastian ends up in the same place as his twin sister, by chance, having no idea that she is alive or that she is pretending to be a man.  When Cesario/Viola is dueling with Sir Andrew, Antonio steps in.  He feels that helping Sebastian is the right thing to do, since Sebastian is his friend.  Then he gets arrested for his past deeds.
Antonio asks Sebastian for his purse.  Cesario refuses.  Antonio is very upset.  He has been an excellent friend to Sebastian, and in his time of need, Sebastian turns on him and refuses to give him his own money!  Cesario/Viola is confused, offering to give what little money he/she has.

ANTONIO
I must entreat of you some of that money.
VIOLA
What money, sir?...
ANTONIO
Will you deny me now?
Is't possible that my deserts to youCan lack persuasion? … (Act 3, Scene 4)

In the end, they get it straightened out that these two are twins, and that Antonio actually gave his purse to one twin and asked for it back from the other.  However, he was still able to offer a valuable service to Viola when he intervened for her in the duel, thinking he was helping Sebastian.  Therefore, he was a friend to both twins.