A bleed is an intentional printing error in which ink extends past the edge of a page, resulting in a margin that appears white. This is done to give the illusion that the printed image continues beyond the edge of the paper.
There are two types of bleeds: full and partial. Full bleeds extend the ink all the way to the edge of the paper, while partial bleeds only extend partially to the edge of the paper. Partial bleeds are more common, as they are less likely to result in an unwanted white border around your printed piece.
The purpose of a bleed is to give your printed piece a professional look by giving it clean, finished edges. Bleeds also make it easier for you to trim your prints down to their final size, as you can simply trim off any excess ink around the edges.
If you are planning on printing a piece with bleeds, be sure to use thicker paper stock (100 lb or higher) so that it can handle the extra ink without warping or buckling. You will also need to leave enough room around your artwork for each side that has a bleed; typically .125 - .25. Your printer should be able to provide you with more specific instructions on how much bleed they require.