Rotating a vehicle's tires is essential to prevent uneven tire wear. Un-rotated tires create increased road noise, lower fuel economy, and reduced wet-weather traction, and usually have to be replaced sooner.
Generally on front-drive vehicles (assuming all tires are the same size), you rotate the front tires to the rear in a straight line, and switch left to right when you move the back tires to the front. In a rear-drive vehicle, you rotate the backs to the front in a straight line and switch left to right when you move the front tires to the back.
On all-wheel or four-wheel-drive vehicles, you would rotate tires in a simple "X." The left front and right rear swap places, and the right front and left rear swap places.
Many sports cars and some luxury and sport-utility vehicles have unidirectional tires, where the tread patterns are designed to perform only in the direction denoted on the tire sidewall. They should always be rotated front to rear to ensure that the direction of revolution does not change.
If you are rotating a full-size spare into the mix, the common practice is to put that tire in the right rear. Check your Owner's Manual for the correct tire-rotation procedure for their vehicle.