Close-up photography--also known as macrophotography--is one of the most exciting things I do with my digital camera. I love the way my camera gives me the ability to grab a slice of the microscopic world and see things from the perspective of an insect.
Macro Photo Basics
Macrophotography is all about capturing the microscopic world--magnifying tiny details that the human eye usually doesn't see. You'll need to set your camera to its macro mode for that. The macro setting is almost universally identified by an icon that looks like a flower, and it rearranges the optics in the camera so that you can get very close to your subject. While you should check your camera's user guide for details, macro modes usually allow you to get to within just a few inches of the subject. In general, the closer you get, the more magnified the picture will be.
At the same time, getting that close has two adverse effects on your photos. First, it amplifies the effects of shake and blur. You'll find it's hard to get a rock-solid, steady picture in macro mode, so I recommend mounting the camera on a tripod or monopod, or even resting it on a bean bag to avoid blurring the picture.
Close-ups also reduce the depth of field, so not very much of the picture will be in sharp focus. You'll need to make sure that you focus specifically on the most important part of the image, since parts of the scene just a fraction of an inch in front or behind will probably be blurry. You can increase the depth of field by turning off auto exposure and setting your camera's aperture to its biggest f-stop, like f/16 or f/22.
Another important consideration is how you plan to orient the subject in the frame. For example, if your subject is long and narrow, like a caterpillar or a tube of toothpaste, try to keep it parallel to the camera lens. That way, the entire length of the subject will stay in focus. If you shoot with the subject pointed toward or away from the lens, only part of it will be in sharp focus.
Mind the Background
Macro photos look best when you can reduce the distractions caused by cluttered, blurry backgrounds. When you shoot outdoors, taking pictures of flowers or insects, perhaps, try using the flash. The background will end up quite dark, eliminating the problem.
If you're taking a picture of something indoors--like a stamp collection or a small gadget that you want to sell on EBay, perhaps--then invest in a sheet of poster board to use as a background. I have a collection of inexpensive poster boards that I keep tucked behind my desk. When I want to take a photo of something very small, I position a board on a table near a window, place the subject it, and shoot. Using a dark colour is great for making the subject stand out in the frame.
Finally, it's worth noting that the flash built into many digital cameras isn't up to the task of illuminating close-up subjects. Often, the flash will overpower your subject or cast a distracting shadow (since the lens may block the flash when you're too close). Instead, consider using a flash that's off the camera.
As an alternative, use a reflector to bounce light from the sun or another light source onto the subject. You can buy an inexpensive reflector from most camera stores, but an even more affordable option is to use a piece of bright white poster board. I have used a square of poster board to reflect light onto my macro subjects with good results.
The bottom line, of course, is to grab your camera and go experiment.