A console is a window where a player / server administrator can input commands to change various aspects of the game / server. Players can access the console with the ` key (usually next to Escape), or by clicking the console logo in the bottom left corner in main menu. Servers administrators don't have to do anything - the server executable is a console.
Using the console
A player can use the console by typing a command followed by arguments (if there are any) and hitting Enter. A full list of console commands with descriptions and arguments can be found here.
Not every command can be executed / have its value changed. Some commands can only be run as a server operator, some commands require cheats to be on, and some are write protected. More information is available here.