When the light from your flash enters your subject's eyes it bounces back from the retina. The retina is rich with blood vessels and colors the returning light red. If your flash is close to the lens (as are many on camera flashes) that light bounces right back into the lens and the eyes appear red. The farther away from the lens you move your flash the more you decrease the chance of red-eye. So the best way to fix red-eye is to get your flash off of your camera. I think more people are beginning to realize this.
However, if you happen to have some shots with red-eye, Lightroom provides a good red-eye removal tool in the Develop module.
Click and the tool will seek and destroy the red. Now if you put your cursor over the red-eye circle it turns into a hand. This allows you to move the circle and better align the removal area. If you hear a warning tone when you first click it means that the tool did not sense enough red to perform the removal. Try clicking a different part of the eye.
If the tool leaves some red fringe you can adjust the size of the removal area. Bring you cursor near the edge of the circle and it will change to a bar with outward point arrows. Click and drag to alter the size and shape of the removal area.
Lightroom's red-eye removal tool provides two sliders to let you fine tune the removal: Pupil Size and Darken. Sometimes when the redeye is removed the pupil will appear light or ghostly. Use the sliders to adjust the size and darkness of the pupil. This will enhance the effect and provide for a more natural looking eye.