Make sure you have good lighting so you can use a fast shutter speed, set your camera to continuous shooting, you may also want to put it on shutter priority.
Also get down on the animals level, or you could start with when the cat is sleeping.
I struggle with mine too. With my cat, I tend to get someone to distract her so that she is sitting still - otherwise she pads over to me and gets really interested in the camera. With my gerbils and hamsters, I quite often give them something to eat to keep them still, and then there's a small window of opportunity when they have finished eating and are starting to move around again, but not too quickly, or they sit there and have a little bit of a wash. I do tend to keep the flash on with the smaller animals, as they seem to be utterly oblivious to it, and I keep a fast, continuous shutter speed.
I was photographing a dog the other day which went mental every time the flash went off to search for focus, so I had to switch to manual focussing for that, but I'm a newbie and don't use it all that often.
For me, it is lots of patience. I sometimes have to take lots of shots to get one good one. At other times, it is really easy to get a good shot. I think lighting and camera angles are the answer for me. I also like to brace myself on something, if I am going to be standing still for a while. If I don't, I get blurry pictures from camera shake.
I tend to use a tripod to reduce camera shake indoors, but I always have the camera within reaching distance set to silent mode. I find the beeps and clicks however quiet ALWAYS gets the pets attention!
I have to take a bajillion pictures to get the few that I like. My cats are quite uncooperative. For every picture I've taken and posted under the tag scm_cats, I've deleted hundreds of really bad blurry shots.
Well I don't consider myself to be in the same league as these animal shooters (and I'm hopeless with the dog) but for the cat my top tip is to get her by the window as the natural lighting is vital and gives a pretty effect coming from one side - apart from that do as the others say - take LOADS and keep the good ones.
My dog is camera shy. She runs away as soon as she sees my camera or my phone! It's crazy. Only one of my cats lets me get photos. He even takes a lot of patience though. If you want to see some really good cat shots look at @shutrfly. Maybe they could give you some tips!
I try to work off my cat's personalities. One cat loves to pose for the camera, so I just catch her in a "posing" mood. The other one is pretty skittish and so I've had better luck photographing her asleep. My parent's kitty is quite playful, so for me patience is the key for her (since my camera is not great)
Make sure you have good lighting so you can use a fast shutter speed, set your camera to continuous shooting, you may also want to put it on shutter priority.
Also get down on the animals level, or you could start with when the cat is sleeping.
@starfish I think my cat has got used to the camera in his face, but I often use the long end of my zoom to get him...such as this...
which also blurs the background nicely. He was sitting on a fence post when I took this, probably watching the birds....
I was photographing a dog the other day which went mental every time the flash went off to search for focus, so I had to switch to manual focussing for that, but I'm a newbie and don't use it all that often.
If you persevere with it you will get some great shots!
It took me about 10 attempts to get this shot!
Bribery! And a soft warm place, and lots of pats
And if none of that works a long lens and lots of patience
But I do realize that it takes a different sort of cat to tolerate this. :-)