Console commands in TF2 change how you play the game, plain and simple. Some modify how your screen looks; for example, hud_saytexttime changes how long chat text is displayed on the screen. Others, cl_interp change how your game communicates with the server that you're connected to.
Before you start playing around with these commands, you must first enable the developer's console. You can do this by clicking "Options" on the main menu, and then "Keyboard options," and then selecting "advanced." By default, both the boxes in the advanced menu (enable developer's console and enable quick switch) are unchecked. Take a moment to check them both now.
You can open up the console whenever TF2 is running by pressing the tilda key (` or ~). Commands are typed in normally, and executed when you press enter. There is a syntax (language) to the commands: most of the commands can be thought of as "off" when the value is 0, and "on" when the value is 1. For example, "cl_allowdownload 1" means that the command allowing the client to download custom files is enable. For info on how to make a list of these commands that is executed every time the game is run, see this post. For now, here's the list of commands that will be useful to the average TF2 player!
Credit to valve for compiling a very complete list of these commands; credit to me for thinning it down and rephrasing it!