Credit unions are the American equivalent of building societies. Building societies have a historic emphasis on mortgages (hence their name) but over time have morphed into de facto banks after legislation passed in the 1980s. Credit unions in America have always started out as thrift and lending institutions and so their evolution into what they are today was more natural than how building societies came to be in the UK. However, credit union membership was initially much more restricted.

There is, however, one interesting difference. In the UK, building societies are permitted to “demutualise” and convert themselves into regular limited companies. When this happens, the members receive shares in the new company and it is then operated like any other company. In the United States, this is not allowed. There is no process for a credit union to convert into a for-profit entity.

Credit unions in the US grew a lot once the US federal government started issuing federal credit union charters that had lax membership requirements. Originally, credit union members had to share some common bond. Thus, it was common to find credit unions attached to trade unions or employers or heavily localised (such as to a single farming community). For example, the largest credit union in America, Navy Federal Credit Union, had a strict membership requirement wherein only servicemembers, retired servicemembers, and their family were allowed to be members. Many credit unions today have only nominal membership requirements (e.g. live within 100 miles of a branch or something else to that effect) and thus almost anyone can join.

Edit: I would like to add that credit unions are not a uniquely American phenomenon, nor did they originate from the US, but credit unions are quite popular in the US. Similarly, neither are building societies restricted to the UK.