I have a Nikon Coolpix digital camera. It's a good camera and easy to use. My only problem is that every time I take a picture requiring the flash, my four-year-old has his eyes closed (and I mean every time). Can you suggest a camera that either does not need a flash (not sure if this is possible, although if I use the camera on my mobile no flash is required) or the next best option? I want to be able to take family pictures at Christmas.
The first thing you should do is check whether your camera has a red-eye reduction mode for the flash. If it does (and most do), try turning it off and using the flash in 'normal' or 'auto' mode instead. I think this may solve your problem.
When you use the flash in red-eye reduction mode, the camera strobes the flash a few times before firing it properly and taking the picture. In theory, when your subject looks at the flashing light, their pupils contract, and this reduces the amount of red-eye -- scary red pupils caused by light reflecting off the back of the eye. In practice, sticking a flashing light in your son's face gives him a chance to close his eyes before you take the picture.
If turning off the red-eye reduction doesn't help, you need to find a camera that can cope with low light, so you can take photos without using the flash. There are two options here, although only one will work well in your situation.
The first option is a camera with image stabilisation. These cameras make it easier to take pictures in low light by compensating for any movement of your hands, so you can hold the camera steady for longer. An image-stabilised camera would be a good option if you wanted to take a picture of your son sleeping, however it won't compensate for any movements your son makes -- which means it isn't the right answer. (Why am I mentioning image stabilisation at all, then? Because if you go into a camera shop and tell them you want to take pictures in low light, they'll probably show you an image-stabilised camera).
The second option is a camera that's more sensitive to light than your current model, so you can get away without using the flash. The sensitivity of the sensor is measured using the ISO rating, which is adjustable on digital cameras (on a film camera, it depends on the film you use). Most digital cameras offer a range from around ISO 100 to ISO 400, with the higher number indicating greater sensitivity. However, some go as high as 1,600 or even 3,200. The higher ratings are more common in digital SLRs, although you will occasionally find them in compact cameras.
Your best option would be a digital SLR such as Canon's EOS 400D, combined with a fast lens such as Canon's inexpensive EF 50mm f/1.8 II (a 'fast' lens has a large maximum aperture and is good at gathering light). The other advantage of an SLR is that it's more responsive, so you can take pictures more quickly -- always an advantage when you're photographing children.
If you'd rather have a compact camera, Fujifilm offers high ISOs on some of its compacts. Take a look at the Fujifilm FinePix F30, or its slimmed-down sibling, the FinePix F20.