Introduced in Borderlands 1, Challenges are basically in-game achievements for doing the basic things, like killing a certain number of enemies or dealing a certain amount of elemental damage. This was further enhanced in Borderlands 2 with the introduction of Badass Ranks.
Basically, completing the many Challenges in the game will give you Badass Ranks and once they reach a certain point, you will get a Badass Token. On the Challenges screen, you are able to spend a Badass Token to increase a certain stat, like Gun Damage or Maximum Health (you get a random selection of five each time you spend a Badass Token). Every stat starts off with a 1% increase, then slowly diminishes each time you choose to boost it.