Rob Spoor gave a nice example recently, but I can't seem to find it now. I found this post by David Byron, which has a link to the Java™ tutorials on.
An Action incorporates an ActionListener, as well as a Map which you can put additional details in. See what you can get from the documentation, and come back if you don't understand something.
I am pretty sure it was you, Rob, but couldn't find anything on a search. I did write an example myself at home to try it out, but can't seem to find it. this is what comes from being a two-PC family; I have probably left it at home. I might be able to look later.
The main decision point is whether you want an object that can respond in the same way to events kicked up by several different UI components.
For example, if you have (a) a button, (b) a menu item, and (3) a custom component, and if they are all supposed to bear the same name and do the same thing, then an Action will allow you to define the Title, Icon, Accelerator or Mnemonic, and functionality in just one place.