Great topic, as a "photo geek" (as Dan said) I also am frustrated by the poor quality of many snapshots I see day in and out.
Problem is that people think the camera will do everything. Their vision is selective, the camera has no selectivity.
Over the last year I've been introducing my 16 year old daughter to photography. She's had to learn some basics and has improved tremendously.
Brice, thanks for starting this topic--I realize many people want to weigh in on advanced topics, it's painful to oversimplify a complex subject. Here's my top tips for beginners using a simple point and shoot automatic camera. BTW, most of these I learned from the very first book on photography I ever read, baci in 1977, and I still think is one of the greats: "How to make good pictures" from Kodak.
1. Get in close. Whatever single element that is most interesting should fill the frame. Want to take a picture of a jewelry box? Have it fill the frame--don't shoot from so far away that you include the workbench it's sitting on. Note--every camera has a limit on how close it can focus. It's no use to try to shoot closer than that. A blurry picture is always to be avoided.
2. Make it sharp--this means the center of attention. Auto focus on modern cameras isn't like your eyes, it doesn't know what you want to be in focus. It will often pick something other than what you want in focus. In low light (like in your house or shop, as opposed to outdoors) it can be difficult to get objects both near and far in focus. Usually the camera can do one or the other.
3. Before you press the shutter, look carefully at all four corners of the picture you're about to take. Is the background cluttered? Are you decapitating someone? Sometimes you need to move stuff in the picture, or else move the camera, to remove a distracting background.
4. Avoid deep shadows (and bright highlights). Digital cameras can't handle a lot of contrast well. A bright sunny day at noon is a nice time for many outdoor activities, but photography isn't one of them. Usually your camera can "see" details either in bright light or in shadows, at the expense of the other one. On a sunny day, shoot in the shade (under a tree, porch, umbrella, etc). On a cloudy day, shoot outdoors.
5. To take great pictures of people on a sunny day, have them stand with the sun behind them (but not where your lens will be pointed at the sun). Now they won't squint, but their faces will be in shadow. Force the built-in flash on your camera to be "on" and it will remove the shadows from their faces. Alternatively, have them stand in the shade and snap away.
6. Use your viewfinder. Most experienced photographers don't hold the camera at arm's length and peer at the digital display to compose a shot. For one thing, the camera won't be as steady as it will closer to your face. For another thing, that display is hard to see in sunlight. Use the viewfinder instead.
7. To take a picture of something taller than wide, rotate the camera. Most people pics (unless they are in a group) look better in "portrait" mode (ie camera held vertically) than "landscape" mode (ie camera held horizontally).
8. Get in close.
9. Get in close.
10. Get in close.