Can my router code be improved?
There's not really much to review here, but a few quick items:


This is going to sound weird at first glance, but you have a route() method that takes a path instead of depending directly on the request uri.

A router's job is to route.  By hard coding the $_SERVER['REQUEST_URI'], you've made it limited to one data source.
route($path) would be much more flexible, and it would mean that one object could be used for routing more than 1 time.
Think about test driven design.  How would you test a hard coded $_SERVER['REQUEST_URI']?  (Other than putting a hacky assignment to it somewhere)

Never assume that array indexes exist

If someone goes to index.php or /foo, so on, then $URLElements might not have 2 elements

As it is, your class does nothing useful -- you probably know that though :)

A major improvement would be making it actually do something :)