1) Either keep the shutter open for longer time or increase ISO. Higher ISO can result in noisy image. Most compact cameras have distinct noise for ISO higher than 400. For ISO above 1600, most images are unuable. However if you really want to capture a photo, this ISO can come handy.
2) If shutter is kept slow so enough light can get in, this can result in camera shake in your hands or the subject in front of you can move too. This results in blurry or shaky image. Normally any shutter speed slower than 1/75 second can result in blurry image.
* Take a bit underexposed photos. For indoor and in low light (and many times outdoor too), I use EV (Exposure Value) Compensation on the camera.
Remember one more thing. We are talking about low light or indoor photos. As you might have read in earlier posts, one good alternative is to add some artificial light or use camera flash. Your camera will love this and with more light around, it will be able to take better photos easily. If with flash, the face is washed out, as it often happens with compact cameras, you can use the same Exposure Compensation and ask camera to take less light in. This will avoid washed out photos in flash photography too.
Enjoy your camera. Actually, your camera can take better photos even in tough light if you are willing to help it a bit.