I'm a Nikon user, can't speak for Canon.
Having handled a D40 (my parents') and a D80, here's the difference between the D40/D60 (the D60 is essentially the same, a few more features here and there and more megapixels but otherwise pretty much the same) and the D80:
If you plan to use the camera purely for snapshots, memories, etc. kind of pictures, i.e. a more powerful kind of point-and-shoot, go with the D40 or D60. This kind of user will usually buy the camera, attach the lens included onto the camera, and leave it there for the forseeable future. Such a user rarely even looks into other lenses let alone be bothered by acronyms like APS-C, AF, DX, and AIS since a camera like the D40 represents a large amount of money invested and any kind of after-market lens costs about half of what your camera cost you, and usually more - if anything, such a user will buy a cheap telephoto at his local camera store, as advised by the salesman, that will work on this user's D40 just fine. Such a user does not require fine-control over his photos since he leaves the camera on automatic all the time.
If you think you might expand into photography as a hobby, get the D80. The D80 allows for much easier access to manual settings in order to get special effects that you can only get with a SLR, as well as allowing the user to use a large number of lenses from Nikon's lens backcatalog, greatly increasing the number of photographic options and looks that the user will be able to achieve. If you are in this category, you will want to make sure that you have a few hundred dollars left over to buy an after-market lens or two, if you barely have enough money to cover just the D80 and a kit-lens, then go with the D40 or wait until you have more money saved.
Hope this helps. As others have suggested, hold the cameras in a retail store to see which system you like better, and read camera reviews and user opinions at sites like dpreview.com.