The web site Cacheopedia defines a challenge cache as a puzzle cache that requires geocachers to complete certain tasks (e.g., to find other geocaches meeting certain requirements) before they can log it. Typically, geocachers must send an email to the challenge cache owner providing evidence that they have completed the challenge requirements, and the cache owner replies with the coordinates of the cache.
Chances are, if you've been geocaching for a while, you've probably already met some of the requirements for a number of the various challenge caches that are out there. For example, most states have a "Counties Challenge" cache, requiring cachers to find one cache in every county in the state before the cache owner provides them with the coordinates to the final cache. In California, this cache is known as, "Discovering and Logging California's 58 Counties".
Similarly, many states or regions also have Quad Challenge Caches and DeLorme Challenge Caches. The former requires you to find at least one cache in each USGS Quadrangle Map section in a given area (see, for example, the Bay Area Quadrangle Challenge), while the latter requires you to find one cache within the boundaries of each page of a state's DeLorme Atlas & Gazetteer (see, the Colorado DeLorme Challenge cache).
Here's a screen shot showing the 70-Quad grid from the Bay Area Quadrangle Challenge listed above:
To log this cache, the successful cacher must first find one cache in each of the map grids shown above, before the cache owner will provide him/her with the coordinates to the final cache. Other interesting challenge caches include: