Actually lestermount’s answer is what most people think. You can use that as a real basic answer but it is more complicated than that. Here are two explanations that are little more accurate for everyday use.
1. If a 15 handicap plays against a scratch golfer, he would need 15 strokes to make the match even. He would receive those strokes on the 15 lowest handicap (hardest) holes.

2. A fifteen handicap would be expected to shoot 15 over par on a course with a neutral slope rating. His expected score and course handicap would be calculated on each course using a formula based on the course slope rating. For the record, a scratch golfer’s expected score would differ based on the difficulty (course rating) of the course. As Tiger said yesterday, a good 10 handicap would not have been able to break 100 on today’s Open course.

If you are interested in more detail, you’d have be a golf freak like me. The USGA offers many publications and even seminars on the subject.

The Club at Longview’s Golf Tip: Decrease Your Handicap