GIMPguru.org has a great tutorial on removing "red eye." That technique gives excellent results, and readers may want to consider using it. However, I wanted a simpler technique that did not require any fine tuning to get results. My technique is essentially the same as the alternate method presented at GIMPguru.org, but with a step added at the end.
Zoom in (+) on the eye. Using the Ellipse Select tool (E), select the pupil. Feather the selection (a value of 2 or 3 usually works well).
Open the Channels dialog box (Image:Dialogs/Channels) and deselect the Green and Blue channels, leaving only the Red channel highlighted.
Desaturate the red channel (Image:Layers/Colors/Desaturate).
Open the Channels dialog box (Image:Dialogs/Channels) and select the Green and Blue channels. All three channels should be highlighted.
In the Levels tool (Image:Layer/Colors/Levels), change the gamma to a value of around 0.70.