It is good to have the eyes of the subject sharp and looking at you – but it is terrible to have them red, and if you take your pictures using your camera's built in flash, there is a good chance that the eyes will come out that way. The red is the light reflected from the back of the subject's eye ball, and when you use your camera flash the light travels straight from the flash to the eye ball and back to the lens, to be captured by the sensor.
Part of the problem is that we mostly use flash when the environment is dark, and when the environment is dark the pupil of the eye opens wide to get more light, and the wide-open pupil makes it easier for the flash light to get in and the reflected light to get out. Now, there are a couple of ways of making the pupil smaller, reducing the risk. One is to shine a bright light at it just before taking the picture, and this is what the pre-flash, or red eye reduction feature, does – it makes a quick flash just before the main flash, giving the pupil time to contract. The problem with this is that it can also cause a change in the subject's expression, and if you do a lot of it, or flashing generally, can lead to a change in the subject's temper.
Now , if the room were brighter, the pupils would not be so dilated, so if you turn up the lights in the room there is a better chance of not cpaturing red eye. Of course, if you turn up the lights in the room, you might not need to use flash at all, particularly with the greater range of noise-free ISO levels in modern SLRs. Even without turning up the lights you may still get a reasonable, if slightly noisy, picture, by using higher ISO and no flash.
If you must use flash, use an external one rather than the built-in one on the camera. If you set the external flash on the camera hot shoe, the change in angle is such that the bounced red light will often miss the lens and so is not captured, and you can avoid any chance of it by aiming the flash away from the subject, something not possible with the built-in flash. If indirect flash doesn't give enough light, you can use a diffuser. Some people advocate keeping the flash off the hot shoe, connecting it by cable or wirelessly to the camera, but now you need to use one of your hands to hold the flash and that leaves only one to hold the camera, which is not enough. You could use some sort of bracket, but that makes things very bulky.
There will always be people who tell you not to worry about red eye, as you can fix it afterwards in your favourite photo editor. The problem with this is that the editor doesn't know what colour your subject's eyes were, and while it can satisfactorily change them from red, it has to change them to something, and often that something won't be right.
You might have noticed that there are two tips this week. This is because we have missed a few since the start of the year, and should by now be on Tip 26, so over the next couple of weeks I'll try to get back on schedule.