There are two causes of blurry pictures. First, the subject is out of focus. Second, the subject or camera moves enough between when the shutter opens and closes that the motion is visible.

Stop Motion
The trick is to be able to look at a blurry photo and know the cause. Think about the scene, look at the photo and figure out what's going on.

If objects appear to streak in one direction your problem is the camera moving, the subject moving or both. To steady the camera lean against something, prop the camera up with both hands and scrunch in your elbows. To deal with a moving subject get the subject to hold still if possible. If not, increase the shutter speed. Increasing the shutter speed will contribute to solving both camera shake and a moving subjects.

Most cameras offer a sports mode. This mode favors a fast shutter speed over all else. The camera will open the aperture and/or raise the ISO to fire the shutter faster. When shooting semi-auto use the shutter priority mode (Tv on Canon cameras). You can then control the shutter speed and the camera will open the aperture and raise the ISO as needed to get what it believes is the correct exposure. For full manual shooting experiment to find the right shutter speed to freeze the action then adjust aperture and ISO to get the exposure you want. Leave a little wiggle room. If 1/50s seems to work, shoot a little faster, maybe 1/80s or 1/100s.

Here's a rule of thumb to get in the right ballpark. When shooting hand held use at least 1/focal length for stationary subjects. So with a 50mm lens, shoot at 1/50s or faster. Keep this in mind when using a zoom lens. As the lens zooms in, increase the shutter speed. At 200mm, 1/200s or faster is required.

Get the Focus Right
Once the motion is frozen the subject may still be blurry or fuzzy. Look at what is in focus for a hint about where things went wrong.

If everything is blurry check that auto focus is turned on. It's easy to bump the switch by accident or forget to switch it back after shooting in manual.

The auto focus brains of most digital cameras is surprisingly good. If something in the shot is in focus the camera picked the wrong thing to focus on. When shooting in automatic modes and the camera gets it wrong about the only thing to do is break the rules of composition and put the subject in the middle to force the camera to focus there.

The alternative method is focus and recompose. This topic deserves a discussion all its own and can go a long way toward improving your photography. For now, here's the basics. Pull out the trusty camera manual and figure out how to set the auto focus point manually. Cameras use a variety of focus points, typically 8 or more, to figure out what to focus on. Rather than let the camera choose, make the camera use one point then put that point exactly where you want to focus. The center auto focus point is the easiest to use. That solves the focus problem but creates a composition problem, hence the recompose. Put the auto focus point where the focus should be, press the shutter half way to lock focus then recompose and shoot. Practice until you can do it without thinking.

Using focus and recompose solves the vast majority of incorrect focus problems. The two special cases that remain are limited depth of field and fast moving subjects.

If the desired focus point is in focus but other parts of the image in front of or behind the subject are not in focus the depth of field is too shallow. To bring a deeper slice into focus close the aperture (higher f/stop) to increase the depth of field. Think of it like squinting to see farther. For a more on depth of field, see Understanding Depth of Field.

Getting the focus in the right place can be complicated by a subject in motion. The shutter may be fast enough to freeze a race car but if the car has already moved from the point the camera focused on the car will be out of focus. There are a few strategies to deal with this. Pan the camera to follow the subject (takes practice) or manually focus on a point you know the subject will be, wait, then fire at the right moment (also tricky). If the camera supports it there are auto focus modes that try to deal with this situation. They come in two varieties. Continuous mode continuously tries to figure out what to focus on. This works well for erratic scenes like sporting events or unruly children. These scenes are no time for focus and recompose so turn off manual selection of the auto focus point. Canon calls this "AI Servo" mode. The second mode locks focus then follows that subject. This works for situations in motion where you at least have the opportunity to pick a subject, maybe picking out one person at a sporting event. Canon calls this "AI Focus" mode.

This Shot

The stairs in the Berkeley station BART stop. Freezing the action here was tricky for an unusual reason - it was shot from a moving escalator! I saw the scene while riding up but the chance to take the shot was past. I rode back down and up a few more times until I got it right. I had to bump up the ISO to 1600 and open the aperture to f/5.6 to allow increasing the shutter speed enough. I also leaned forward then moved my body contrary to the direction of the moving escalator (not the safest manuever but it worked).