There are plenty of recipes out there that call for only egg whites or only egg yolks. Angel food cake springs to mind as the most obvious example, since it calls for 10-12 egg whites only, but you can end up with a few leftover after making something as simple as an egg white omelette.
Storing egg whites and egg yolks is simple, though the two egg parts do require different storage methods. Egg whites can be kept covered in the fridge for a few days, but when you have a lot the best thing to do is freeze them. Pour each egg white into one cube of an ice tray and freeze. Transfer to a bag and all you need to do is defrost them at room temperature when they’re called for. Egg yolks don’t freeze too well and can dry out after a day in the fridge alone. The best way to store them is by mixing them with a bit of water first (you can probably keep the yolks whole in the water if you are careful and simply remove them whole when ready to use), before putting them into a covered container in the fridge. They’ll keep for 2-3 days this way.
As an additional note, I sometimes find that I want to store whole eggs. When I make blown Easter eggs, for instance, I usually cook some and store others whole. When doing it this way, I like to mix the yolk and the white together and place in in a covered, refrigerated container. I’ve seen people freeze whole eggs, but I honestly always use them in a day or two and find that they are unaffected by that time in the fridge.