Atoms are Fallout 76's microtransaction currency, earned by completing challenges. The delivery of Atoms begins almost immediately with simple challenges like killing your first creature or picking your first lock, and it continues to trickle in as you fight and forage your way through Fallout 76. You can purchase Atoms with real cash, too.
What's for sale in the Atomic Shop? Mostly cosmetic items: outfits, hats, poses for photo mode, alternate skins for your Pip-Boy and armor, and some items for your camp. There are some emotes that might come in handy, such as expressing that you're hungry or thirsty to other players as well as a number of icons to use in your social profile.
You can see the monocle and cowboy hat I purchased below. Note: some of these items need to be crafted in-game at a workbench once you've purchased them.
Below you'll see what you can currently spend your Atoms on in the Atomic Shop. You can click the upper right corner to enlarge the images. There are bound to be many more items available in the future.