Bleed refers to any element on a page that is printed beyond the edge of the paper. Whenever you see anything (text, graphics, photographs) that is printed right up to the edge of the paper, it was actually printed onto larger paper over the margin guidelines, and the paper was trimmed.
Some press operators might tell you, "No bleeds allowed." Some might say, "We charge extra for bleeds." They might ask, "Does it bleed?" You might say, "Can I bleed this line?" An image that bleeds off all four edges of the paper is called a "full bleed." This page you are reading has no bleeds; for this page size and the press we used, bleeds were not allowed.
The reason the printer might not allow a bleed is because she plans to print the job on the right -size paper, without any trimming. If the image prints right up to the very edge of the paper, then the ink "bleeds" off the paper and gets all over the press and makes a mess.