Google Wave works well as standalone system for chat, file sharing and so on. However, it also gives you the option to add extensions which enhance your experience and the range of functions it can perform. Extensions can be thought of as similar to Facebook applications; just as you can install apps to customise the way you use Facebook, you can add extensions to personalise the way you use Google Wave.
Extensions come in two varieties: gadgets and robots. Gadgets are the most similar to Facebook apps, you install them and they run as part of a wave (thread). To give an example: you can install the Sudoku gadget, and use it to play against a group of friends. Robots are more similar to the bots you see on Twitter, but far more sophisticated because they behave as kind of artificially intelligent participant in a wave. The best example is the Twitter robot, Tweety, which can be used to update your Twitter status and also to follow tweets from other users.