The name “administrator” comes from the Windows world. In the Unix world, “system administrator” is a job description, but “administrator” doesn't mean anything special with respect to accounts.
Unlike Windows, Unix accounts do not intrinsically have a notion of privilege. The privileges in an account are conferred by the files that they can access, by the setuid commands they can run, by the sudo rules they are allowed to use, by the applicable SELinux policies, etc. These can be conferred directly to a user, or to a group that the user belongs to.
There is one exception: the account whose user ID is 0 gets a lot of extra permissions (basically, the permission to do just about everything). This account is conventionally called
root (it would be possible to use another name, as far as the kernel is concerned, since the kernel doesn't know about user names, but that would break a lot of administration-related software).
The root account is not meant for users to log into in normal operation. It's a system account. The administrator only runs commands as root to perform system configuration tasks, not to do other work. The administrator may log in as root sometimes to perform system installations or repairs, but wouldn't use the root account for non-system-related tasks like web browsing and email, they'd use their personal account for that.
Informally speaking, an administrator account is the account of a user who is able to run commands as root. Often this is done through sudo, a program which allows users to run commands as other users (including root) if they are authorized to do so by the sudo configuration.
Privileges are associated with a process and inherited by their child processes. When you run a process with elevated privileges (e.g. through
sudo), only that process and the processes that it starts have elevated privileges. The rest of the session is not affected.