Tracing a family last name is called genealogy and can be done by searching databases, public records and family trees from all over the world. Before 1066, last names were not used in Europe and England, as nicknames were common ways to identify people in communities large enough that a distinction was needed apart from a first name. After 1066, the practice of using last names gradually began to spread after being introduced by Norman barons; the result was that nicknames became last names and could have been taken from a place, a job or a physical attribute.
Tracing a last name and developing a family tree that gives a clear history of where it came from has been made easier with the advent of the Internet and the cataloging of historic documents such as church records, court documents and specialized directories.
An Internet search under genealogy yields pages of results for specific sites that charge a fee to trace a family's roots, but researching the databases that are openly available to the public often are a better place to start.
In the United States, there are searchable databases from government census findings, and while not the exhaustive database of all their holdings, the National Archive´s website has online resources that can help a person start the search for the history of a last name.