The administrator, or admin of a server, is the person who takes responsibility for that server. It may be that the server is running from their machine, or that they simply have jurisdiction over a remote server, or in some cases part of one.
Terminology
The term admin is not interchangeable with operator (or "op"). An op is a person designated, by the original administrator or other ops, to patrol the server on their behalf. They basically are a co-owner at some purposes. Take this responsibility seriously.
Powers
Administrators have all of the commands available to ops, and also the advantage of the tools available from the server software. Admins have control over the selection and implementation of the map, and can choose the settings for the room, such as maximum number of players, players allowed from the same IP address, and the name and MOTD of the room.
The admin can also ghost the room; chatting via the console, without being logged into the server.
On some servers, an admin does not have access to the console. They usually still have access to almost every command, though.