Challenges are small objective based missions that can be played alone or in co-op. Note that in the parlance of Brink, a "Mission" is what we might otherwise simply refer to as a map or match. Challenges are comparable to tutorials in Modern Warfare, they are short intense missions which teach you how to play the game, but you can play them over and over again to try and improve your times.
You can use Challenges to unlock new stuff since everything you do in single player or co-op carries over to multi-player. Each challenge has 3 levels - the 1st one is easy, the 2nd one is a bit more difficult, and the 3rd one "Is a [censored]in' [censored] to complete" (Ham's exact words). The 3rd level is designed to be hard for the developers themselves, but after you complete level 3 then there is a leaderboard to try to work your way up as well.
A note about leaderboards, Splash damage has said there will be no global leaderboards (or at least none based on kills and deaths). So if your whole life revolves around trying to get to the top of a leaderboard, Challenges should make you happy. My theory is that Splash damage has included this leaderboard feature in the challenges to cater to those players who want it, yet eliminate from the regular MP games that type of behavior inherent when players focus to much on a leaderboard or badge which is detrimental to team based play. In other words, if you are busy focusing on some engineering badge, you might be reluctant to play another class when needed until your badge is achieved.