Bleeds allow you to run artwork to the edge of a page. On a press, the artwork is printed on a large sheet of paper and then trimmed down to size. If you do not allow for a 1/8 of an inch bleed, any misalignment while cutting will result with the artwork not running to the edge of the paper. Bleeds ensure you get the results you need.
For example, if you have designed a standard 3.5" x 2" business card with a red background covering the whole area, you will need to enlarge that red background to 3.75" x 2.25". This will make the red background extend 1/8" on every side of the page.
Small mechanical variations can end up leaving a hairline white edge where there should be no white edge at all, if the image is not extended beyond the final trim size. Extending images 1/8" beyond the final trim size guarantees that images truly will go all the way to the edge of the printed paper.
For a more in-depth tutorial on how to set up your document to accomodate for bleed, see "Page Bleeds," an article from PrepressX.com.