Challenges are in-game tasks that can be completed to earn Badass Ranks, and subsequently, Badass Tokens. Also, there are one of the only ways to get customization options for your characters. Challenges usually include things like killing a certain amount of enemies, or other things like using a certain weapon to inflict a certain status effect on an enemy a number of times.
All Challenges, except location-based ones, have 5 levels, which always yield 1/5/10/50/100 Badass Ranks, respectively.