Why don't they use normal air in race car tires?  

Many race car teams use nitrogen instead of air in their tires because nitrogen has a much more consistent rate of expansion and contraction compared to the usual air. Often, a half pound of pressure will radically affect traction and handling. With track and tire temperatures varying over the duration of a race, the consistency of nitrogen is needed.

Nitrogen pressure is more consistent than normal air pressure, because air typically contains varying amounts of moisture due to changes in the relative humidity on race day. Water causes air to be inconsistent in its rate of expansion and contraction. So, a humid race in the southeast United States or a dry race in the desert western United States could make for unpredictable tire pressures if "dry" nitrogen were not used.

Nitrogen is also used in the high-pressure tires on large and small aircraft.