There are three types of irony: dramatic, situational and verbal. Situational irony is when something happens that is not what you expect.
One of the biggest examples of irony does not become entirely clear until later. This is that Severus Snape is charged with protecting Harry even though Harry was the son of his enemy in school. Harry and his friends assume that Snape is out to get Harry. For example, when they see Snape keeping eye contact and mouthing a spell at the Quidditch match, they assume that he and not Quirrel is trying to hurt Harry. Snape is actually trying to protecting Harry.
Dramatic irony is when the reader knows something that the characters do not. In this case, the reader realizes that there must be some kind of history between Snape and Harry, or Snape would not be acting like this. Harry, on the other hand, just assumes that Snape is mean to him and does not really know why.
A final example of irony is Hagrid and Fluffy. Hagrid tells a stranger that the way to tame Fluffy is to use music. He tells this story to the children when they visit him. Quirrel uses what he learns to get past the dog, so the children do not have to use this skill. It is also unusual and unexpected that each task plays to the strengths of one of the children. The potion test, the flying keys, and the chess game are all designed so that one of the children can use his or her ability when the others could not.
Within the context of the series as a whole, great sources of irony include the fact that Severus Snape was a super secret double agent. Time and time and time again, Harry Potter suspects Severus Snape of being in league with Voldemort. Yet, at the very end we know that he was truly loyal.