Monads are very useful in computer science, specifically logic programming. Haskell is a programming language which is extremely mathematical, to the point where programs and proofs are the same thing. It's a pure functional programming language, which means everything is a function and these functions are not allowed to have side-effects (you can't cheat to make Haskell do something for you). So it should not come as a surprise that ideas from category theory lead to new ways to do things in Haskell. Monads are used in just this way, to get Haskell to do more things than you can get without monads. For example, recursive things. Monads are so useful that someone has website containing a big list of things known to be monads. I saw this website in a talk given by Emilio Gallego, a student of Jim Lipton. I believe there is a link to it from here.
The discussion of monads on wikipedia is very good. You can see all the terms from categorical monads popping up in that article, and facts about monads from category theory often lead to new programs in Haskell and other programming languages. Another nice overview is this pdf. An article written for computer scientists is here.
If someone with more expertise comes along and wants to add to this answer, please do.