What will cause the right signal light on the front not to work but the rear signal light works?

Typically, a turn signal that works on one end of the car, but not the other, is just a burned bulb. It's when one whole side works and the other doesn't that things get a little more involved. Some European cars will alert the driver when a bulb burns out, and temporarily disable that circuit until the bulb is replaced and the key cycled on to off and back to on.