Archaeology is a study of human past based on material evidence.
Just like historians work with written sources, archaeologists work with material remains, be it past societies' material culture (such as buildings, graves, weapons, pottery, jewelry, etc) or environmental data humans left behind. A material remain in archaeology can be a man-made artifact, or a human remain itself (skeleton, for example).
Understanding ecology and environmental conditions past societies face is also important, and that's why archaeologists often analyze animal bones (to tell which animals people eat and used), as well as botanical remains (to tell which plants they eat but also what kind of climate people faced.
The purpose is reconstructing human past: the way people lived, their societies, their beliefs. Contrary to the popular belief, archaeology is NOT about searching for beautiful objects, such as gold vessels or ancient jewelry. A simple ceramic pot, a bone or a coprolite* is often more useful than any beautiful crown or an ancient sword made with rubies. So finding beautiful artifacts and putting them in a museum is not the purpose of archaeology.
*Coprolites are fossilized feces. They can come from a human or an animals, and they are really valuable because they give a lot of information on what individual in question ate, in which way food was prepared, as well as a health condition of said individual. So they are very, very valuable finds.