The letter Feste speaks of and Fabian reads is the one Malvolio wrote while imprisoned, saying that he has been wronged by Olivia and that he has her own letter to prove it.
The letter Malvolio has is the one Maria wrote pretending to be Olivia when she and Fabian and Sir Toby were setting up Malvolio to think Olivia was head over heels in love with him.
When Malvolio gets this letter, he does as he is instructed (he thinks by Olivia). The instructions are funny in that they ask Malvolio, usually darkly conservative, to behave outrageously. As a result, he is deemed out of his mind and is bound and placed in a dark room. When he is finally freed from the room, he chastizes Oliva, saying she has "done him wrong" by instructing him to behave one way and then imprisoning him for it. He produces this letter as evidence. She takes one look at the letter and says that it is not her handwriting, but Maria's.