When your body burns calories and you lose weight, obviously mass is leaving your body.  In what form does it leave?  In other words, what is the physical process by which the body loses weight when it burns its fuel?  

Somebody said it leaves the body in the form of heat but I knew this is wrong, since heat is simply the internal kinetic energy of a lump of matter and doesn't have anything do with mass.  Obviously the chemical reactions going on in the body cause it to produce heat, but this alone won't reduce its mass.

There's a lot of detail you could go into with regard to this question, as is done in the other answers and comments, but I think the answer itself is pretty simple. Imagine a surface that just barely surrounds your body, as if you shrink-wrapped a body in plastic. By the law of conservation of mass (valid in non-relativistic physics), the only way your body can lose any amount of mass is for that amount of mass to pass out through the surface. So you just have to consider what bodily functions cause that to happen. I think they've all been identified in the comments:


Exhaling
Sweating
Excretion (in the nontechnical sense of, roughly, things you do in the bathroom)


Actually, any dead skin cells, strands of hair, etc. that fall off you would also count, although my guess is that those represent a minor contribution.

As a bonus, the "shrink-wrap view" also makes it easy to identify the ways in which you gain mass, by looking for all processes that cause matter to be drawn in through the invisible surface:


Eating & drinking - solids and liquids through the esophagus and gastrointestinal tract
Inhaling - gas through the trachea and lungs


The thing is, when most people talk about losing weight, they're referring to a long-term average loss of mass, which means that the processes in the first list have to remove more mass over some extended period of time than the ones in the second list bring in. This clearly requires some of the preexisting mass in your body to be converted into the waste forms that you can dispose of through excretion, exhaling, and sweating. This preexisting mass generally tends to be body fat. The other answers do a pretty good job filling in the details of how the fat gets converted to waste products.
