Challenges are a constantly changing series of tasks, the completion of which earns bonus credits. There are two kinds of challenges: one Weekly Challenge, starting on Mondays and generally requiring a larger amount of effort over the course of a week, and four or five Daily Challenges, generally (though not always) offering fewer credits for smaller tasks. The cutoff time for challenges is 3am PST.
Challenges are a way for Bungie to encourage a broad experience of their game, to entice players to not just play Multiplayer, for instance. One day the challenges may be more weighted towards Firefight, while the next has a split between Campaign and Multiplayer or reward the use of a specific weapon.
Any challenge requiring Firefight can be performed solo via Score Attack. This not only allows you to take on all the enemies yourself (requiring fewer games for a kill goal) but also gives the opportunity, every time, to play Gruntpocalypse, which arms you with a DMR against waves of easily headshottable Grunts, leading to sprees and other Medals that give a large credit boost. --Sigafoos
Just a note: the challenges URL describes them as 'global challenges' -- I wonder if that means they plan on providing other types in the future? I don't know what would be more 'local,' unless I could issue them to my friends. --Sigafoos