Levequests are quests given to you in most major cities and some camps around Eorzea, given my levemates.
Levequests can range depending on your class, if you are a combat class, they generally involving killing a certain number/type of monster. If you are a gatherer, you usually need to collecting a certain amount of ore/vegetation/etc. If you're a crafter, you are tasked with making an item.
You can only accept a certain number of levequests, 16 at a time, and as long as you have 'Allowance'. Allowance is needed to accept levequests and you earn it automatically, at 3 a day I believe (you can hold 100 max).
You can view the current number of levequests completed under your 'Achievements' tab in the menu.