A somatic mutation is a mutation that occurs in the somatic (i.e. non-reproductive) cells of an organism; this is in contrast to a germinal mutation, one that occurs within the reproductive cells. When we think of a mutation, we are usually thinking of germinal mutation, that is, a mutation that can be passed on to offspring, thereby driving evolution. Somatic mutations cannot be passed onto gametes.
Somatic cells do not play a part in reproduction of offspring, but they do divide when the organism grows. Thus, a somatic mutation in one cell can spread, as the cell divides, to become a significant portion of the organism. The classic example of this would be a flower petal or leaf that has one cell mutated to display a different color; as the cell divides, a wedge-shaped splotch of the new color will appear on the growing leaf. Somatic mutations that occur early in development will have larger effects than those that appear later in development. In the leaf example, this would mean a larger or smaller wedge.
Along the same lines, cancer is often caused by a by a somatic mutation which reproduces itself in the body (the most common cancer causing mutation is simply that the cells start to reproduce themselves more often than they usually would). Other somatic mutations that appear in humans include mosaicism of various sorts and having of eyes of two different colors (or sometimes just a section of the iris of one eye in a different color).
While somatic mutations cannot be reproduced sexually, in plants they can be spread through budding or grafting by humans; seedless grapes, the Washington Navel orange (although not other types of seedless oranges), and bananas are all somatic mutations that have been kept going by humans.
Our immune systems are also dependant on a specific type of controlled somatic mutation. When B-cells are first formed in the bone marrow, they don't have any information on what antigens they will be attacking. They travel to germinal centers in the lymph nodes, where they encounter specific antigens. When a B-cell encounters one of these prototypical antigens, it triggers a mutation in the gene sequence that controls the formation of the immunoglobulin (Ig) on the surface of the B-cell. These in turn are the 'keyholes' into which various antigens will fit and be recognized.
As with any type of mutation, many somatic mutations pass unnoticed, either because they have no phenotypically apparent effect, because they appear so late in development that they affect only a few cells, or because they took place in junk DNA, with no detectable change to the appearance or functioning of the organism.