SuperTuxKart
Or STK for short, is one of the most popular racing games available freely and open-source on Linux. You can find install it in any major distribution with the default package manager. SuperTuxKart comes with several game modes, like single racing or Story mode, vehicles models, inspired by mascots in the free software world (you can drive with Tux, FreeBSD devil, GNU etc) and a whole bunch of maps, which you can unlock by completing missions in the story mode. STK also includes weapons to slow down your opponents. There is an in-game add-on menu from which you can download even further maps and models.

TORCS
It stands for The Open Racing Car Simulator, and it is a free racing game. It can be ran in fullscreen or windowed mode, and it supports different resolutions. TORCS can be a little confusing at first, since the camera will focus another car during a race. To race in first-person mode, press F2.

Stunt Rally
Stunt Rally is a 3D rally game with modern graphics and lots of features. There are dozens of tracks to race on, taking place in various environments, 19 car models and also an online multiplayer mode. The OpenGL graphics are really wonderful, but it also needs a decent dedicated graphics card to play at an acceptable frame rate with high settings. You can install and play Stunt Rally on Desura.

Speed Dreams
Being a fork of TORCS, Speed Racer is “an open motorsport simulator” with various graphics options, several tracks and car models. You can install Speed Dreams from playdeb.net, the official PPA or on Desura. To install Speed Dreams from the PPA, open a terminal and type:

Dolphinity Racer
Yet another free racing game for Linux striving for graphical realism.