There are two primary kinds of blur in photos (well, three, but we’ll assume you keep your camera fairly clean): focus blur and motion blur. Here are examples of focus and motion blur, respectively (click to enlarge):
Focus blur happens when the subject of your photo is simply out of focus. The solution to that is to make sure your autofocus is on and try again. If it’s out of focus, re-focus and shoot again. Pretty straightforward. On point and shoot cameras, the most likely reason you’re out of focus is because the subject moved or the smart focusing system wasn’t so smart and focused on the wrong object.
Motion blur, on the other hand, doesn’t happen because your subject is out of focus. It happens because your subject is moving relative to the camera frame while the exposure is being made AND the shutter speed isn’t fast enough to freeze it. Let’s tackle those two aspects separately.
First: motion. Motion in the frame can happen in three ways: either the subject is moving or the camera is moving or both. Imagine taking a photo of cars on a highway (just not in California—you’d have no trouble getting sharp photos there). Now imagine taking photos of people at a bus stop from inside a moving car. Those are examples of two ways motion can be introduced into a photo. Camera shake is also an ever present problem caused by the motion of your hands and arms while taking a photo.
Motion by itself, however, isn’t enough. You must also have a slow enough shutter speed for the motion to become visible in the exposure. When the shutter speed is slow (below 1/60 of a second), even relatively slow motions show up in photos. And slow shutter speeds are usually caused by insufficient light. That’s why you rarely see motion blur problems outdoors on sunny days.
So, the solution is to increase your shutter speed. And often the only way to do that is to add more light. One obvious way to do that is to use your flash. If you’re inside during the daytime, you could also just go outdoors. Sometimes the difference between shade and sun is all the extra light you need.
You can also increase shutter speed by decreasing (widening) your aperture. A wider aperture lets in more light giving you faster shutter speeds. If you are at your widest aperture and you still aren’t getting enough speed… well, now you know why people pay through the nose for so-called “fast” lenses: glass with f/stops 2.8 or wider. (You can also try a faster ISO setting.)
What are good shutter speeds for freezing action? It depends on the apparent speed of the subject through the frame. What do I mean by “apparent speed?” Imagine a car 10 miles away from you moving at 60 miles per hour across the horizon. From that distance, it’s moving very slowly. But stand right next to it and it whizzes by much faster. Here are some minimum rule-of-thumb shutter speeds that I go by:
People posing for portraits: 1/30s
People walking: 1/120s
Pets and children: 1/250s
Closeup sports/action: 1/500s
But remember, every situation is going to be a little different. Shooting a basketball game from the sideline is a different animal than shooting one from row 20.