In Information Security theory a process, or more abstractly the user invoking that process, has access to some subset of all the information and the functions available in a computing system. This is considered the privilege level. This is an abstract concept, not something defined such as "privilege levels 1-10". For example, a user may need to read files from directory A and write a log to directory B, but not change or delete files in directory A or any other directory. Another user that can read and write to these files & directories and other directories has a higher level of privilege.
For example, on Unix based systems the root user (user id 0) has full access to everything and therefore full privilege. On Windows NT based systems, the kernel has full privilege. Users that can't install programs, load printer drivers and so forth have far smaller privilege set.
The principle of least privilege states that users and processes should only have access to the least amount of information and functions required to do their job. Why this is preferable will become clear.