Cameras love light.
How people perceive light and how cameras do is quite different. While people adjust organically to varying degrees of illumination so everything appears 'normal' - cameras take a more absolute approach.
What does this mean? While you might not notice the difference between looking around your room inside, and looking out a window - the camera does.
For this post, I'm mainly focusing on Point and Shoot cameras - as opposed to SLR - but much of the information is universally applicable.
How do cameras compensate? They do one (or more) of a few things:
If there isn't enough light, the camera makes more. Trouble is, this only works for a few feet away and it eats up the battery power. It also makes a obvious impact on the quality of the picture - it's very easy to tell when a flash is used on a point and shoot camera.
When you prevent the flash from going off in low light, the camera has other ways to adjust. The image sensor in the camera can be set to require less brightness to register a viewable image. Trade-offs for this include more noise (skittles!) in the picture and at higher settings a less natural looking shot - flatter colours and not as much variation. The highest ISO setting usually has to be set manually for these reasons. It's often better to have a sharp, albeit noisy and flat, shot than a blurry one.
If neither of the above are used - or it hits the upper limit on ISO, the camera has no choice but to keep the shutter open longer. This almost always leads to a blurry shot. You can set a 2s timer on most cameras and that will let you brace the camera better (rather than pressing the shutter) and/or (preferably and) use a tripod. Not a lot of situations lend themselves to those two tricks though.
So what else can you do?
Outside, even on a cloudy day, has significantly more light than you will find inside. If you're inside - more lights than seem necessary will really help your shots. Stand near windows, with your back to them, if you can help it. Don't blame the camera.