Without seeing an example image, it's hard to tell which of the following is the problem.
- Your lens is on manual focus.
Note that having the camera's exposure dial on "A" is not the same thing as auto-focus. Auto-focus is usually controlled on the side of the lens itself with a "M/A" switch.
- The subject is moving too fast for your shutter speed.
If you are photographing a person, use a shutter speed of 1/250s or faster. If you are photographing a fast-moving person such as a kid, animal or sports, use 1/500s or faster - that is, if you want to eliminate motion blur.
If you cannot get to this shutter speed, you probably need more light or a lens that can open to a wider aperture.
- The camera is shaking.
If your shutter speed is too slow, even the small movement of your hands on the camera can cause blur due to camera shake. As a general rule, avoid using a shutter speed slower than 1/50s, or if you are zoomed in (telephoto), avoid using a shutter speed slower than 1/(focal length), ie 1/135s if you are zoomed in to 135mm.
If you cannot get to this shutter speed, you probably need more light or a lens that can open to a wider aperture.
- Subject is moving out of focus.
If the subject is moving, then they may have moved out of focus in between the time the autofocus mechanism focused, and the camera took the photo. This is particularly true if you are very close to the subject and they are moving fairly fast toward or away from you.
Try to get your subject still, or use a wider shot. Some cameras are faster at focusing and exposing than others.
- You haven't focused on the subject.
Cameras will only focus on a certain part of the image at once. In full auto mode, it usually picks an area to focus intelligently, which works most of the time but sometimes picks the wrong part of the image to focus on.
The quickest fix to this is to half-press the shutter to focus while directly pointing at the subject you most want to focus on (most cameras give emphasis to what's in the middle of the frame), keep it half-pressed, and then move the camera to take the picture you originally wanted, then fully-press the shutter. You can also set the camera to choose a specific focus area instead of choosing one automatically, which will make this technique even more effective (and necessary).
- The subject is too close.
The lens has a limit to how close you can take photos. This is probably only going to be the case if you are extremely close, say less than 12 inches from the subject on a typical lens. Some lenses can focus very close, say 4cm (2 inches). Others designed for viewing distant objects may have trouble below 2 feet.
Normally if this is the case, there will be a light or beeping noise (or different/lack of beeping noise) to indicate the autofocus failed.
- There is not enough light for the autofocus to work well.
In dark environments such as a dimly lit room, the autofocus sensors in a DSLR may not work as well, may work more slowly, or may not work at all. Even in situations where there is enough light to take a photo (say, using a high ISO or using a flash) there still may not be enough light for the autofocus sensors to work well.
Many cameras have an autofocus-assist light (orange or red light coming on during autofocus) which you can enable to help with this in some situations. Again, the camera should show a light, or beep (or not beep/beep differently) to indicate that the autofocus is having problems.
Bonus hint: some cameras have an autofocus sensor in the centre of the frame that is better able to cope with low light than the surrounding autofocus sensors. Try lining up your subject in the centre of the frame on half-press, if you're in a dark environment.