Community manager is the name given to someone who does Community management. It is most commonly used as a job title for professional CMs.
A community manager's job may encompass many specific roles and activities in relation to the community, including:
The community manager of a commercially-led project will typically be responsible for reporting to his management on the state of community, the management of initiatives to grow the community, and the maintenance of community processes. She may also take a lead role in product development to ensure that internal teams are also following these processes.
The primary concern of a community manager is to ensure that the conditions are created for a community to grow. That means documenting how a newcomer can get involved, ensuring that paid developers do not live in an ivory tower or work in isolation from the community, creating and maintaining mentorship programs, ensuring the availability of any community infrastructure which might help the community to become autonomous, intervening as required in community conflicts, and in general maintaining a smoothly operating transparent and productive work environment for all involved.
This allegory (at least, I think it's an allegory) of different management approaches lends itself well to community management. You can be all hustle and bustle, you can think of your community in a paternalistic way, or you can simply observe where pain points arrive and resolve them as they do. Many of the pain points will be human in nature, rather than technological.