Analogous colors lie next to each other on the color wheel.
A comfortable and pleasing color combination, analogous colors are often found in nature. Think about leaves during the fall as they change from green to yellow to orange to red. They're going through an analogous color scheme.
Typically one of the analogs will be a dominating color, one supporting, and the third as an accent.
To calculate the analogs of a primary color you rotate the color wheel the same distance in both directions. For example, on this site we display analogs that are 30° apart. There's nothing magical about the 30° measurement. It's just what we went with. The important thing is that the center, primary color is the same distance from the colors on either side.
For an example, take a look at the color wheel on this page. Blue on the far left is the primary color. The other two colors are the same distance from blue and are analogs of blue.