Fortunately, you are not alone. This is a very common issue with most cameras, including consumer SLR cameras.
If you have noticed, most of the time, this happens when you are taking photos in Auto mode specially in low light. A camera needs some fixed amount of light to take a good photo and the light enters through the lens on the camera and gets captured by its sensor. There are 3 things that a camera can control to get the enough amount of light it needs.
1) Shutter speed: When you click on the Shutter Release button on the camera, the shutter opens up for some amount of time to let the light pass through the lens. Your camera decides the speed of the shutter based on many factors. If you are taking a photo in a low light, the camera would need to keep the shutter open for longer time but Camera knows two important restictions. (i) First one is the shake in you hands. Most individuals can't hold a camera or anything as such, perfectly steady for more than 1/75 seconds. Surprised? Please don't. A minor shake in your hand will cause the photo to be blurry. (ii) Even if you have steady hands, or you use a tripod, most of the time, the thing you taking a photo of will move. Most of the time, we take photos of people. They also move during a fraction of the second. A kid or a pet will move more quickly then an adult but we all constantly move. Our naked eyes may not see the move but a camera lens will capture it. If you are taking a photo of a landscape which has a tree, a minor breeze of air will cause the leaves to move. In short, photographer as well as most subjects move quickly, most cameras will try to avoid shutter speed slower than say 1/75 second. This is a restriction imposed by camera's software to save your photos in most situations by avoiding chances for blurr.
2) Lens Opening: The technical term is Aperture. Most compact cameras have tiny lens openings. The wider the opening, the more light a camera can take. However most sub $1000 camera's have ordinarly lenses which do not allow too much light to pass through. No surprise a fast lens sometimes costs much more than a best compact camera you can buy. In short, for this Lens Opening, your camera doesn't have move room to play around. This is a camera's hardware restriction.
3) Sensor's sensitivity: The technical term is ISO. When a camera runs into the first two limitations (of limited shutter speed and lens opening), the only option it has is to increase the camera's sensitivity to light. This is like increasing sensor's receptivity to light. Camera can play up to some level of higher ISO but the more ISO or sensitivity it increasing, the image starts getting noise. The noise is the unwanted pixels or grainy look in the photos. If you want more tech details about noise, see the Wiki post : http://en.wikipedia.org/wiki/Image_noise
For most compact cameras, when the ISO is set to more than 400 or 800, you will start seeing the noise.
Now having learned about the 3 main things a camera can control to take a nice photo for you, if you see your camera take a bad photo, it is probably not your camera's fault. This is similar to many limitations we run into our daily lives. I like to run a marathon but I know I can't run more than 3 miles. I like to make a million dollars a year but I know most of us can't. Same way, you and your camera wants to take a great photo but many times it can't. So what do we do to take better photos?
photos in better light:
* If possible take photos in good light- natural or artificial. Make it easy for your camera to help you out.
* If lighting is dull when you are taking photos, set camera ISO to 100 (200 or 400 at most) and add more artificial light. Or, simply turn on Camera flash. Use the camera in Fill Flash mode. You will get some shadows or washed out faces but that would be better than ghost looking blurry photos. In the next article, I will show you how to avoid shadows or over-exposure too.
Tripod when taking photos of stationary or slow-moving objects:
* Use Tripod and set Camera to Av Mode with ISO fixed at 100, 200 or 400 only. If you are taking photos of people or pets, use the smallest F number on the camera. If you are taking a photo of a landscape, set it to 8 or so. In Av mode, you decide the ISO and Aperture and let camera decide the shutter speed. Because you are using a tripod and the you are taking photos of a stationary object, camera can use a slow shutter speed and can shutter open as much as it needs. You will see nice beautiful non-grainy photos.
expensive cameras or a fast lens:
* However use of a tripod or slow shutter doesn't work when you are taking photos of people or pets. They usually move within a split second and that causes photos to get blurry with slower shutter speed. To overcome this too, use some faster lens like 1.4, 1.8 or 2. This will help camera take lot more light while shutter is open. Also, more expensive camera's have bigger sensors and a bigger sensor does help with better control of noise. Also an SLR can take less grainy photos with ISO as high as 1600 but a regular compact camera would have noisy images when ISO starts getting above say 400.