The study of Archaeology in the west more or less began as "antiquarianism" -- an interest in ancient artifacts -- sculpture, art, and vases -- that itself developed from a renewed interest in ancient literature as it was rediscovered from the late Medieval/early Renaissance periods -- that is, from the 14th-15th centuries.
For years -- until the 18th century -- this material was largely of interest for the manner in which it could be used to support moral and ethical views of government and social behavior; and artists were attracted to it for both aesthetic and narrative/illustrative reasons. There weren't systematic attempts to place objects in context and to grasp exactly how they had been created and understood in their own times and places.
That changed in the 18th century, partly as a result of the uncovering of Pompeii and Herculaneum. This is a very long and complicated story, but it did lead to modern, scientific techniques in archaeological excavations and studies.
The Wiki site on "Archaeology" isn't great, but it offers some useful links:
https://en.wikipedia.org/wiki/Archaeology 
