Noise is an unfortunate reality of photography. It was a problem for film, and it's a problem for digital cameras. The more sensitive the film or sensor is, the more prone it is to picking up unwanted variations in an image. Bits of film grain, funky pixels, annoying artifacts, they all tend to pop up when you crank up the ISO sensitivity.
Of course, you probably don't have $7,000 to drop on a pro-grade body/lens combination for low-light shots. Fortunately, there are several things you can do that don't cost extra money. A flash can provide enough light to shoot at lower ISO levels, and that means less noise. However, flashes can often blow out subjects, making them look flat and ghostly. You can offset this by setting your camera's white balance to "Flash" mode, or by shooting in RAW and using Photoshop or another RAW processing program to alter the white balance later.
Neat Image is a Photoshop plug-in (also available as a 32-bit standalone Windows application) that can process noisy images and filter out the grain, specks, and other artifacts that come from shooting at high ISO settings. Imagenomic Noiseware is a similar, stand-alone program that can also filter out noise in photos.
It's an unfortunate truth that noise just comes with the territory of photography. You can do a few different things to keep it to a minimum, but it's not completely unavoidable.
Photo credit: Flickr user Tom@HK