Taking good pictures takes a bit of practice [some luck doesnt hurt]. There's a certain skill in taking good pictures. The most important thing about taking better pictures is to practice things that will allow you to take better pictures. The items below are a few important tips (some obvious, some not) to get you from being someone that takes pictures that bore people to tears to someone that takes enjoyable photographs.

Use the Focus Lock: Here's something that I'm sure has happened to you. You take a picture of 2 people standing side by side. Your automatic focus camera decides to focus on the wall 10 feet behind your 2 friends -- blurring the picture of your friends. Most cameras, even point-and-shoot cameras, have a feature called focus lock. Your cameras automatic settings will be set (focus, aperture and/or shutter speed) will get set when you push the button 1/2 way down. This allows you to move the camera and retain the focused distance and exposure on your subject. The next time you take a picture of 2 people, try the following:

Center the 2 people you're going to take a picture of. This will make sure that the layout is okay

Move the camera (turn) to 1 of the 2 people.

Press the shutter down 1/2 way. This will focus the camera on that person, getting the distance correct, but dont change the zoom setting.

Move the camera (turn) back to your original layout in #1 above.

Take the picture.

Get Close, Don't take a bad snap-shot: Here's a popular one. A lot of people take pictures of people (these are called, snapshots). A lot of picture takers are afraid to move in close to the people in the picture (the subject). There's a lot of things can go wrong with your picture when you do this (lighting, poor focus, subject gets lost in a sea of scenery, etc.). Zoom in close to the subject of your picture, let the subject take up most of the frame. Just recently my brother took a picture of my wife, my daughter and I after an event. We took up the center of the picture with a huge border around us (which actually took up most of the frame). The picture would have come out so much better if he zoomed in on us and forgot about the background, instead I had to chop away more than 1/2 of the picture to get the results that I wanted.

Don't get too close: When you are taking portraits of people with a 35mm camera, the optimal focal length of your lens should be 85-135mm. The reason for this is two-fold. First, if you use less than 85mm, you'll need to stand too close to the subject and you'll end up distorting their face (typically this is seen by having a persons perceived HUGE nose). The reason that you don't want to use more than 135mm is because you'll distort the person by flatteing ou their features, making them become part of the background.

Construct Better Lighting: Understand the lighting of your picture. Many many pictures are taken of a person (or persons) outside on a bright sunny day. Your automatic camera is designed to compensate for either shadows or bright spots -- but not both in the same picture. The automatic exposure in your camera is designed for 18% gray -- this means that an average picture is considered to be 18% covered (when converted to grayscale). You can see evidence of this by taking a picture of snow -- you'll note that it almost always looks gray when the camera determines all of the settings. The easiest way to adjust your exposure is to make sure that you are focusing on the bright highlighted areas in the picture (use the focus lock). When you focus on the highlights, you will usually wash out the color in the sky, but at least the persons face won't be blacked out with a shadow.

Read the instructions: This one's my favorite. When showing people how to use features of cameras that they've purchased, a typical response that I get is 'oh, I didn't know you could do that!'. In my head, I'm thinking, well why not, it was on page 24 of your instruction manual -- that's where I found it. Reading through your instruction manual [with the camera in your hand] will do wonders to understanding what you can and can't do when you're taking pictures.

Use a Tripod: This isn't always the easiest thing in the world to do. If you're just taking snapshots and don't feel like lugging one around or if that's just not your thing, then you can't. If you CAN use a tripod, you can really cut down on what's known as camera shake. There's a lot of different things that can cause a little bit of shake, a tripod will help you keep the camera still. Camera shake shows up as some blurriness in the picture, it will typically look as though the camera was moving up and down (which it was).

Take LOTS of Pictures: This one may or may not be so obvious. The more pictures that you take the more likely you are to get better at constructing a good shot, you'll also get a better understanding of your camera and how to use it. Make sure that you review the pictures that you've taken with a critical eye for both picture-taking technique as well as photo subject and layout. Without doing this last step you'll never know what kind of progress you're making.

See also:
Photo Tips
Software