The blurring is due to a shutter speed that is too slow. If you are shooting on 'full auto', the flash should fire to compensate for this. If you have disabled the flash, that would explain everything. I'm not a big fan of flash photography myself, since it fills your picture with harsh light (unless you know what you're doing, and set your flash power lower, etc.).
Actually, the full answer to your question is that there isn't enough light coming into the lens, to allow a high enough shutter speed - and by that I mean a shutter speed that is fast enough to 'freeze' any movement. This speed is related to your subject - if you are photographing someone that is sitting still on a couch somewhere, the shutter speed generally needs to be faster than 1/30. However, this is also related to your zoom level (the longer the lens, the easier you get 'camera shake' - which is also a form of blur, but caused by your own inability to hold the camera still enough). As a rule of thumb, your shutter speed needs to be roughly the same as your lens length in order to cut out camera shake - so if you are shooting on a 300mm lens, your shutterspeed should be at least 1/300, but preferably a bit faster. On the other end, your subject dictates how much actual blur you'll get if you have moving subjects in your shot. A man walking his dog will have significantly less blur as a fighter jet screaming through your frame - even at exactly the same shutterspeed and lens length.
So how do you make sure that you get enough light in in order to get a high enough shutter speed? Well, you have several options:
1. Add flash (yuck) (and ineffective if your subject is too far away, like on a soccer pitch if you are in the stands)
2. Increase your ISO - by increasing the sensitivity of your sensor, you effectively make it do more with less light. This can, however, lead to excessive noise in many cameras, and CCDs are particularly prone to noise. I wouldn't push it beyond 400 ISO for general use
3. Change your shooting mode to "Tv" as opposed to Auto, and choose a shutterspeed that is in keeping with your lens length and subject. The camera will then (try to) choose a suitable F-Stop on your behalf
This last option won't allow any more light to reach the sensor per se, but it will give you control over the shutterspeed, and if you control it well enough, your problem will go away. Essentially this is the same as the sports mode, though the sports mode usually also changes the type of autofocus from single shot to servo mode, or even AI mode on some cameras.
I sincerely hope I haven't confused you with all of this!
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