Sensors in digital cameras are divided into tiny light-sensitive cells. Each cell contributes to the contents of a single pixel. For example, some cameras have cells sensitive to red, green, and blue light. The output from these cells are combined to produce a single RGB pixel.
The cells are so small that random noise becomes a significant factor. The smaller the cell, the more obvious the noise. This is why professional DSLR cameras, with large sensors, have less noise then consumer cameras with small sensors. Sensitivity also affects noise levels. As you increase ISO from 100 to 400, for example, you are effectively turning up the volume and noise levels increase.
To minimize noise, turn down in-camera sharpening and sharpen as the last step in your workflow. Be sure to specify a non-zero Threshold in the Unsharp Mask dialog box. Otherwise you'll re-instantiate that noise that you worked so hard to hide!
You can also reduce noise through post-processing with special software such as NoiseWare, Noise Ninja and Neat Image. Michael Almond has a nice web site that compares several products. The following figure is a 100% crop of an ISO1600 image has been partially corrected with Neat Image.