A bleed is the area around the finished piece size that the artwork extends into.  The bleed will be cut off the piece, but allowing the artwork to extend outside of the final dimensions ensures that there is no white space on the border of the piece.  The bleed also ensures that design irregularities, small paper shifts during printing, and cutting inconsistencies do not affect the look of the final printed piece.  Generally a 1/8” bleed is recommended for color printing.  

Bleeds are necessary because most printers cannot print over the edge of a piece of paper.  Thus, if you attempt to print full-color artwork on paper that has the same dimensions as the final artwork size there will be a white border around the piece.  Bleeds ensure that there is no border around the piece and the artwork extends all the way to the edge of the finished piece.
When designing artwork with bleeds it is important to include crop marks which will indicate to the printer where the piece should be cut.  It is also recommendable to design the artwork so that there is a safe zone 1/8” inside the crop marks that does not contain text.  This safe zone will ensure that not text is removed during the cutting process.

Adding bleeds is essential when printing artwork where artwork extends to the edge of the page, but when graphics and text are contained within the piece it is better to print without bleeds.