It is the system message log file. It holds kernal messages and messages from the applications.
A kernel is the central part of an operating system. It manages the tasks of the computer and the hardware - most notably memory and CPU time.
Example: When you run some software or script, before it executes completely to reach the final result. It goes through so many processes. If it get failed somewhere in the middle after any process, message till that process are saved in a file as log so that the user can interpret where exactly the sofware is having problem.