Taxidermy.
Laurie at The Itty Bitty Kitty Committee is an absolute *master *at posing kittens. Nobody has any idea how she does it! Some of us suspects she glues them down. 
But seriously, her kitten pictures are adorable, and the blog is great too.
I think I fibbed a bit further up: ONE of my cats is terribly photogenic. But going through some older pictures, I can see that my little girl is the one who’s the big challenge. Her brother is longer-haired and different patterned and somehow…just soREGAL. (Until he runs into the wall. He’s really a great big dork.)
She…well, she’s a different story…I don’t know if it’s cause she’s so fluffy or what, but I can never get a good one of her. So I tend to just catch her sleeping.
Sometimes you are better off just waiting until they’re sleeping, or sleepy.
Serious squeees happening now. What wonderful shots. I wonder if she could be cloned to come and help out here.
I can’t claim to be an expert photographer, but I’m here if we’re showing off our kitties!
Sometimes, the picture perfect moments just happen, and you take the picture with whatever is available. In this case, I used my iPhone camera. (This is Charlie, going on an adventure through the azaleas. We also have an awesome picture of him going on an adventure through the Christmas tree, but I just have that on film.)
Sometimes, the glowy eye effect from the flash is kind of cool. (This is Linus, who is apparently Buddhist.)
Olives, this picture was taken with a Nikon on the portrait setting, which your camera probably has. It’s the one with the icon that looks like a woman’s head. Pretty sure it’s the wide aperture thing people are talking about above. (This picture is also Linus.)
StarsApart - what beautiful pictures of your kitties!
Shhh! It’s a secret! 
Which Nikon Coolpix?
Just keep shooting. It’s digital. You don’t have to buy a new roll of film every 36 pics.
keep shooting! It will be good eventually 
I’ve had some luck with both my dog and cats by getting close while they’re asleep, getting the camera ready, and then calling their name to wake them up, and snapping the picture.
They’re usually calm enough that the photo looks like they’re just sitting there relaxing and looking at the camera.
This might not work for every pet, and some might think of this as not being very nice to the cat, but I’ve heard of placing a basket or bucket upside down over the cat until it settles down, prepare the camera, and then remove the bucket and take the picture.
Of course your cat has souls - he keeps them in the basement, and steals more from people whenever he can… 
Si