I'd appreciate some help here from US or UK folks (especially UK).

Is there a name for an exam that a person has to take from some state body to be able to work in a certain profession. I often hear reference in the US to a "bar" in the context of law.

The context I need it in is a person passed this test for being a librarian. Is there a name specifically for that?

No... not a single term. But there are professional bodies for which levels of membership may have certain criteria associated. 

For example I am a member of the institute of physics. To get the level of member I needed a supporter and to provide evidence of working at a given level as a professional phycisist for a given amount of time. 

Many of these bodies provide chartership as well as membership grades, e.g. chartered phycisist or chartered engineer.