A developer console is a text-based input system to execute commands and change variables in a program.
Start your Steam client and open the library. Right-click on the Brink entry in the list and select "Properties". Press the button labeled "Set launch options...". In the following window, enter "+set com_allowConsole 1"(without the quotes) and press OK.
After a restart of the game, the console can be accessed ingame by pressing the "`"-key (the key left of the "1"-key).
Open the console with the previously enabled keybind. You can now execute various commands by entering their name and corresponding arguments and then pressing enter. To get a description of the commands, enter the command "listCmds".
You can also change the console variables by either issuing the command "set [variable] [new value]" or by entering "[variable]=[new value]" and then pressing enter. Most variables are cheat-protected, which means they can't be changed unless you started a map with the command "devmap". A list of all unprotected variables can be found here.