Question: Why do those red eyes appear?
Answer: When strong white light enters human eye, some of it reflects, and most of the visible reflected light is red. So, when you shoot with flash, chances are people on your photos will have red eyes. Voila - Demon eyes made easy

Question: How can I prevent this ?
Answer: Here is one simple trick that sometimes can do the magic.

The light enters our eyes through our pupils. Pupils actually act as eyes apertures - they open and close, allowing more or less light to enter. Tell your subject to look at something bright for a short period of time (be careful. Don't make them look at the sun or similar dangerous objects for our eyes). Their pupils will automatically close in order to lower the amount of light entering the eye. Now, tell them to look at you and BOOM! flash them out! Since pupils were smaller, less light from your flash entered the eye, so less chance to see a red eye (d)effect.
Some photo cameras have no-red-eye mode which works in similar way: it flashes few times, which closes subject's pupils, and then the photo is made.
This trick, while useful, is not always working.

Question: How to really prevent red eyes in a photo?
Answer: Imagine that you are standing in front of a mirror, and you have flashlight in your hand. (Or, if you want, do it for real with lights off! Ideal flashlights for this experiment are the ones which have focused light. Do not try this with lasers! While they are ideal for experiment, they are also ideal for ruining your eyes!)
Put the flashlight in front of your eyebrows, pointing exactly at reflection of your eyes at the mirror. The mirror reflects the light and the light will strongly shine in your eyes.
Now, take flashlight to your right (as far as your hand goes). Point its light at the same place in the mirror.
You will notice that the light that shines directly at your eyes is none.
This is something that we are familiar with.
Now, if your eyes were camera sensor, flashlight was your portable flash unit, and the point in the mirror you were lighting was your photo subject - it should be clear that:

In order to prevent red eyes you should have your light source moved further from your lens.
How much further depends on how big light source is, how strong it is, how far you are from the subject etc.

Have fun flashing your subjects