Dramatic irony occurs when audiences know what characters in a book or play do not. In act I, we as an audience know that the "male" "Cesario" is actually Viola, a woman, something that everyone in the play but Viola is unaware of. We also know another fact that the characters do not, which is that Viola/Cesario is in love with Duke Orsino.
This dramatic irony creates situational irony, which occurs when a situation is the opposite of what it seems to be. Viola seems to be a man wooing Olivia for the duke. In reality, she is a woman who wishes she could marry the duke. Ironically, she is trapped in the situation of trying to succeed in a task—getting Olivia to marry Orsino—that she hopes with all her heart will fail. As Viola/Cesario puts it:
Whoe'er I woo, myself would be his wife.
Verbal irony is when words means the opposite of they appear on the surface to mean. An example of verbal irony occurs in act I, scene 3, when Sir Andrew asks Sir Toby about his hair, saying:
Tt becomes me well enough, does ’t not?
When Sir Toby replies that Sir Andrew's hair is "excellent," he is being ironic. In fact, he thinks it looks like so bad he hopes someone will give Sir Andrew syphilis so that he goes bald.