A user is generally the human on whose behalf a particular computation is being done. On a time-sharing system, it would be the person logged in to the system over a terminal (real or virtual). On a personal computer, it would be the person using that PC.
The term is sometimes used to refer to a process (or similar locus of computation) which is doing things on behalf of, or at the direction of, the human.
On machines which have hardware support for time-sharing, the CPU usually has two modes, 'User' and 'Kernel'; in the former certain instructions which can disturb the overall operation for other users are prohibited, and only allowed to the operating system, which runs in the latter mode.