Archaeology is the scientific study of important physical remnants of the cultural past for the purpose of better understanding human history.
In order to really comprehend what archaeology is all about, you must examine all of the key words in the definition and also note what the definition doesn't say. Archaeology requires a scientific approach, it deals with physical remnants, it studies the cultural past, and it is about understanding this past, not just describing it.
It does not say archaeology is about digging, although that is often an important archaeological activity. It does not say archaeology is only about artifacts, although those are the most common physical remains of the past. It does not say that archaeology only deals with the ancient past, although the older physical remnants are, the more interesting they usually become to most archaeologists and to the general public.
Scientific study means archaeology uses a rigorous, formal, systematic method. This method helps us to better find archaeological sites, properly analyze archaeological remains, better understand what we find, and adequately summarize what we have learned. Physical remnants include artifacts found on the surface and those excavated from below the surface, as well as ruined structures and recognizable human disturbances of the landscape (features) such as pits dug in the soil.
The cultural past involves only the time period of humans, which includes our distant ancestors of the genus Homo who began making stone tools in Africa over two million years ago. Because archaeologists only study the human past, we do not study dinosaurs. The last of the dinosaurs disappeared over 60 million years ago; paleontologists study dinosaurs.
Finally, archaeological analysis is not an end in itself, but must include public conclusions about the results of the study. Science must be broadly shared or it is of limited use.