Functional programming is another programming paradigm that is popular, mostly in academics. The best example of a functional programming language is Haskell and Standard ML.
The fundamental difference between functional programming and object oriented programming is that you are programming in the sense of data flow instead of control flow. See the presentation Taming Effects with Functional Programming by Simon Peyton-Jones for a good introduction.
A good example of functional programming used in the industry is Erlang. It is mostly used in telecommunication, distributed and fault tolerant systems. See the presentation Erlang – Software for a concurrent World by Joe Armstrong.
There are also newer functional programming languages that combine functional programming with OOP. Two good examples are F# for the .NET platform and Scala for the Java platform; they can often use existing libraries on the platform written in other languages.
The trend of new programming languages now is Multi-paradigm, where multiple paradigms like object oriented programming and functional programming are combined in the same language.