There are three types of challenges in Doom Eternal.
Starting with Cultist Base, and ending with Urdak, each map offer three special objectives (such as finding an item, or killing a certain type of demon in a specific way). Successfully completing all three challenges grants an extra Sentinel battery upon completing the map.
After maxing-out a weapon's upgrades, a special challenge for that weapon can be attempted. Successfully completing it unlocks mastery for this weapon. In the late game, mastery tokens can be found hidden in maps, they allow to unlock weapon mastery without needing to complete the challenge.
Mastery tokens are found in:
Weekly challenges are organized as part of the Slayers Club and require being connected to one's Bethesda.net account. As the name implies, they are changed on a weekly basis. Completing them gives bonus XP which are used to increase rank and unlock cosmetic items.