Red Eye is caused by the dilation of the pupils of the eye. If the room is dark and you take a photo with flash, the light enters the pupil before it has a chance to contract. You will see this more with people that have lighter colored eyes. Turn on more ambient light to try to remedy this.
It's not a problem with the phone's camera per se. You will get this with any real camera.
Otherwise what
natediddy1120 said - mess with the settings.
You also need to understand that some stuff may not be able to be "fixed" with the camera in the phone. It obviously doesn't have the best lens, and not a large sensor for the system. Those two things are major drawbacks to quality. It's low-light qualities are quite lacking.
That being said, I've gotten some acceptable images as long as there's enough light. And if I really wanted to take photos, I'd use one of my real cameras I own - not a phone.