Photographing kids part one: babies
First things first, why separate baby photography from photographing kids in general? Well, with babies you've got all the time in the world to 'create' your photo. Their world is still very small, so you can move that around to any location that suits you best (i.e. having good light and an uncluttered background). Babies sleep a lot, and when they are awake they are not able to move around yet. This gives you plenty of time to set up your composition, and you can even use a tripod very effectively! Try this with those older kids that are running around like mad and you now why this is so different...
Take advantage of natural light as much as possible

It's not that I'm a particular flash hater; to the contrary! Flash can be very helpful if used appropriately. But for baby pictures I really like the somewhat "softer mood" that natural light creates. It always hurts my eyes when I see all these baby pictures where this has been blasted away with (onboard) flash...
To take full advantage of natural light, move your kid close to a window, but just outside direct sunlight to avoid harsh shadows. Make sure that at least part of the light is directed towards the face.
If you want to hand hold your camera check that there's enough light to give you a shutter speed of at least 1/80th. If it's too dark you can always use a tripod (if your baby is at rest).
So this means that you should never use a flash? No, but try to use it as a fill flash only and not as the dominant light source. Fill flash can help you to create a catch light in the eyes of the baby (window light can be great for this too) or to enhance the skin tones by lighting it up just a little. Using the aperture or night scene mode of your camera normally activates fill flash.
...and for those candid opportunities in less ideal light, by all means use your flash as you normally do to grab that moment!
For portraits use a short tele lens
Most baby pictures are portrait photos and on top of that you probably want to create a flattering portrait too. I mean: Have you ever seen a caring parent that said, "Hey, have a look at my baby's big ears and nose!"
You can achieve such a portrait by taking advantage of the compression of perspective that a short tele lens provides (or zoom in to around 80 to 90 mm). You can use a longer focal length too, but you probably will not have enough room indoors and handholding your camera will be much more critical.
An additional note about focal length. The 80 to 90 mm is meant for a 35mm camera. If you have digital slr you probably have a 1.5 or 1.6 focal length multiplier. This means that a focal length of 50mm will be around 80mm effectively. The rule of thumb for a digital compact camera is to zoom in to around 2 to 3 times.
Use a shallow depth of field to create that dreamy mood
You probably already have admired those baby photos where only the baby itself or (parts of) the face are in focus. This effect is achieved by taking advantage of depth of field. Depth of field refers to the areas of the photograph both in front and behind the main focus point, which remain sharp. For portrait photos, and in my opinion, even more so for most baby portraits, you want a shallow depth of field to create a three dimensional effect with that 'dreamy mood'.
Depth of field is influenced by a combination of aperture, subject distance and focal length. Since we already determined the focal length (and because of this the subject distance), all we have left is the aperture. Use an aperture that is as large as possible (at least f2.8 but preferably even larger).
A word of warning; When photographing with a small depth of field, focussing becomes very critical. If you don't pay attention to this you'll end up with a sharp nose or ears instead of the eyes. For apertures greater than F2.0 the auto focus normally isn't accurate enough and you should use manual focus.
Note: Do you know that the cheap 50mm prime lenses are great portrait lenses for digital slr cameras with the 1.5 or 1.6 multiplication factor? They become around 80mm effectively and have a maximum aperture of F1.8, which is great for achieving a minimal depth of field.
If you use a compact digital camera, achieving a shallow depth of field is 'problematic' to say the least. Zooming in as much as possible might help but don't set your hopes too high. This is one of the main reasons I'm not much of a fan of these cameras...
Deal with unnecessary background clutter
I already mentioned that you can move your baby around easily. Use this 'feature' of your kid to deal with unnecessary background clutter. You don't need a uniform background per se and it's fine to have baby toys etc. around, but look out for backgrounds that mainly consist of wires, tables, chairs and body parts of other people. Another common trap is to have a background that is brighter than your subject, like artificial lights, a sunlit window or a television that is switched on. If you can't get an uncluttered background take advantage of the shallow depth of field to throw the background out of focus.
The right composition and timing are what make the photograph

So far I've mainly talked about technical issues that just help to create the conditions for the actual picture taking.
We are now at the moment where the all-important creativity starts to take place. The right composition and timing are what 'make' the photograph.
Think about what you want to achieve: A sleeping beauty, a baby hugged by mum, having fun with dad etc. etc. Be patient and wait for that right moment to happen instead of blasting away with the shutter button.
I don't think writing pages of words will help you here. The best learning tool is to look at other people's photos and get inspired by them. Look around in magazines and the many photos you can find on the web, and hopefully, you can get some inspiration from the samples on this page too!
Alfred