First, remember that irony is often a subset of sarcasm (that is, you can be ironic and sarcastic at the same time). Essentiallu, sarcasm is any cutting remark, while irony is a remark that attempts to make a point by saying the opposite of what is meant.
For example, let's say you've just had an argument with someone, and you said something that made the other person leave the room in tears. I was watching the whole thing. If I want to be sarcastic, I might say, 'Well, that wasn't a very good thing to say, was it?' But if I'm being ironic, I might say, 'What a marvellous way with words you have!'
Of course, if I say the latter, I mean the exact opposite of what I say, which is the essence of irony. Note, though, that the irony is still a cutting remark (i.e. sarcastic). Hope that clears things up!