Removing the red eye effect can only be done by posing a change somehow, so the light from the flash doesn’t bounce back into the lens. This can be done several ways:
(a)Turn off the flashlight. This is the point most people forget about. Obviously, if you can do without a flash, either by increasing the light in the room, by switching to a film that has a faster ISO value*, or using a faster lens**
(b)Diffuse the flashlight. If you hold a piece of (white) cloth in front of the flashlight, you will notice that the intensity of the flashlight is only a little decreased. It does, however, seem less piercing to your eyes. This is not a coincidence - if you hold a piece of thin white cloth in front of your flash, you can often drastically decrease the problem with red eyes, because the light gets diffused and “bounces” around in the room before it hits your subject.
(c)Move the flashlight further away from your eyes This is a bit harder with compact cameras, but if you have an SLR, you should definitely get an external flash that goes in the hot shoe of your camera. For one thing, these flashes are a lot more sophisticated than the internal ones, but they are also are significantly further away from the lens.
(d)You could try to turn on the anti-red-eye function that probably exists on your camera. This function usually sends off a few short flashes, or it will shine some other sharp light into your “victim’s” eyes. This makes their irises smaller, and the problem diminishes.
(e)Turn on more lights. This has the same effect as above, but it also has some other advantages: One, you get more even light, two, depending how sophisticated your camera is, it might fire a less strong flash, giving a more natural light. Three: you might get away without using a flash altogether.