There are a few types of Challenges, all found through your Options menu. On this screen, press Right on your D-Pad to bring up Daily, Weekly, as well as other classifications of Challenges that will offer you either ATOMs, S.C.O.R.E., or even Perk Card Packs if you play Nuclear Winter mode more often. Challenges can range anywhere from hacking computers, picking locks, reaching specific levels, or discovering locations. Best to check some of these before you try, and there has been some instances where it doesn't immediately unlock, if at all.
Typical challenges can consist of:
Levelling up to 4, 6, 8, 10 (and then incremental levels of 5 after 10 to 100).
Picking, equipping, and then ranking up a Perk Card in each slot.
Collecting caps while in specific location/regions (Ash Heap, The Forest, The Mire, Savage Divide, Toxic Valley, Cranberry Bog)
Collecting collectibles (Vault Boys Bobbleheads and Magazines)
Acquiring specific diseases once
Collecting resources, cooking, crafting, scrapping, harvesting