You can take great photographs of your cat by following a few simple guidelines.

It's easier to take a nice sharp photo than it was when photography was a
rather new invention. Exposures took several minutes, and a cat could not
hold still for that long! The photo on the left was taken in the 1860's, more
than 20 years after the invention of photography. This cat must have been
very special, because pets were not photographed often. One would have to
go to a professional studio for a portrait.
This photograph was taken in the early 1900's. A great idea is to get in the picture with your cat! Have someone photograph the two of you interacting. It is very evident this lady loves her cat. Notice how they are both looking in the same direction.

It's a good idea to get at the same level as the cat's eyes. If your cat is on a chair, get down to her level. Professional photographers use this rule for the most flattering and traditional poses for humans, so it would work the same for a cat.

If you want to get your cat at an unusual angle, your photo will have a more "artsy" quality.

Watch your cat throughout the day, and notice what it is that makes you smile. It might be when your cat is sleeping, or playing. Then try to get a few good shots of that activity. Try different angles, and pay attention to whatever is in the background. You don't want an object to appear as if it where sticking out from the cat's head!
When photographing a black cat, you want to get as much detail as possible. Try to keep any large white or light colored areas to a minimum. These areas tend to throw too much light back at at the camera, and the black cat will lack detail as a result. The photo to the left has just the right amount of white, and because it surrounds the cat, sets the cat off even better. The rest of the photo consists of midtone colors, so the baby and the cat stand out as the main subjects.

To the right is our great cat, Daphne. She's basking in a patch of sunlight on the floor, so her head is well lit. Even though her eyes are almost washed out with the bright sun, I love the way Daphne's face shows all it's features. By getting all the way down on the floor in front of her, I was able to make her the main focus of the photo.
Speaking of sunlight, I think it is the best light source for photographing a cat, that is, if you're not a studio photographer with great lighting equipment. It's a soft, natural light that shows the cat's truest colors.

Phoebe, left, was taken with natural light coming through a window.

Flash photography usually flattens out a cat's features, and often casts a hard shadow behind the cat. Sometimes, bouncing a flash off the ceiling will help soften it's harshness. Also, a flash aimed at a cat while she's looking at you will bounce back off the inner eye surface and make her have "spooky eyes," just as people get "red eye" for the same reason.

Phoebe (left) and Penny (right) in different lighting situations. Both are photographed using sunlight coming through a window. But Penny is in direct sun, and Phoebe is not. Notice how much softer the shadow areas are in Phoebe's photo. When taking a photo of a cat in direct sun, if possible, hold a white piece of cardboard or any light colored paper up to the side of the cat that will be in shadow. This will help bounce some light back onto the dark side of the cat, so its features will show up.