If you own a point and shoot camera and usually only use the auto mode please take some time and read this. If you own a DSLR and only use the auto mode, then you really need to read this. Not due to elitist reasons, but simply because you can often get a much better picture by not using it. The auto mode, as the name implies uses the sensor the camera comes equipped with to decide exposure time, f-number(aperture size), iso, white balance, if flash should be used, what to keep in focus and if focus need to be changed.
Needless to say the algorithms doesn't always get everything right, and while some things are harder to set manually than others(such as focus on point and shoot cameras) there are some settings that can make the difference between a good and a bad photo. And in some situations where you have time to change settings between multiple photos you can with some tinkering greatly improve the outcome of the photos.
The general points you should consider, other than keeping your object of interest in frame is the following:
Exposure is how long the sensor is exposed for every photo, the longer the exposure the more light is gathered, but movements become blurred. Unless your camera is mounted on a tripod or resting on a solid surface and your motive is largely immobile you should try to keep this as low as possible. I usually don't change this manually.
F-number/aperture (f/5,6 for example) This is the size of the aperture, a bigger aperture(which translates to a lower f-number) lets in more light, which results in lower exposure times. A large aperture also makes the depth of focus more narrow which gives Bokeh(blurred background/foreground effect). A personal preference is to use Av(aperture priority) mode with the lowest f-number possible at almost all times.
ISO is the light sensitivity, it is however a double edged sword, because it introduces noise while lowering exposure time, learn how to change this because you'll want to do this often. Knowing your camera and what ISO settings introduce excessive amounts of noise is useful. You'll want to avoid blurring due to high exposures, while keeping noise down to an acceptable level.
Flash is the most misused of camera features. Sure it lets you take blur free photos in low light with low ISO, but the artificial and harsh nature of the flash most often makes for a bad picture. The exception is hot-shoe flash with diffuser or celing bounce which can be used for a much more plesant flash experience, though it's a bit beyond basic photography . What you should do is find the flash setting, and either use flash off always, or flash on when you need flash. Auto flash is notoriously trigger happy from my experience, and it can be mildly embarassing to have it trigger in a relatively bright room when you're trying to take a picture discretely. It also drains battery. I use flash a last measure when i want to take a photo but neither aperture, ISO or solid support can allow for a good picture.
White balance also known as colour balance, it is one is the settings which i usually find that auto takes care of quite well. It differs a bit between the brands and models though, my pentax k-x have a slight bias towards yellow for example. I usually leaves this to auto as it's one of the details that can be corrected quite decently after the photo is taken, and if you shoot in raw you can change this just as well as when the photo was taken.
AutoFocus is usually the best choice for focusing, both on point and shoot cameras and DSLRs. The exception is very low light situations where the AF sensor is essentially blind. It can also be useful if you know the position of an event, say a bird in a particular tree, you can then use the AF to get the tree in focus and switch to manual focus(or use AF-lock setting) and get your pictures without the AF searching for focus and delaying your shots or trying to focus on something mundane behind your object of interest, say a cloud behind that fighter jet passing over your head. Most if not all cameras also have settings where you can select the general size of the autofocus target and where it should look for it in the frame, allowing you to autofocus on small objects in the right upper corner instead of large central objects.
That should be the main points, there might be more, or less, depending on personal preference. I would advice anyone with a camera to not just find these settings and investigate them but to play around with any and all settings you can find, you might find something useful, or you might just get familiar with the settings, both which will help you take a better photo.