How does mass leave the body when you lose weight? 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.
 A: Here is a credible article that explains how MUCH of weight loss occurs through exhaling CO2
http://en.wikipedia.org/wiki/User:Znode/Misc/The_Physics_of_Losing_Weight
Plus, we loose mass through our intestines as the liver filters the blood in the form of bile.
http://www.liverdoctor.com/liver-problems/weight-loss/
And we obviously lose water through kidney function.
A: Essentially, losing of weight occurs by means of burning fuels precisely like your car does when it burns petrol and emits exhaust gases.
The only difference is that for humans that fuel is to be found in the form of sugars. The fat is what you want to get ultimately rid off, of course, but sugars are more easily processed and so this is what you are removing first.
The basic aerobic cycle is the Krebs cycle.

But to reach it first, glucose needs to be broken down first to pyruvate (by an anaerobic process of glycolysis) and then to acetyl-CoA. One can gain some energy from this reduction but not much. The real energy is hiding in the actual Krebs cycle but for it one needs (besides lots of other stuff) the mentioned acetyl-CoA and oxygen (this explains why you don't get enough energy when not breathing properly) producing the carbon dioxide and some energy that is stored in the $ATP$ (adenosine-triphosphate) and transported to wherever it is needed inside the cell (you are mostly interested in muscle contractions performed by muscle cells). So, you'll burn whatever amount of sugar you have ready in the body. You'll also lose carbon (initially stored in the glucose) by exhaling in the form of $CO_2$. There is also additional hydrogen produced and carried away in the form of $NADH$ and $FADH_2$. It's hard to estimate where it will end up though, as it is (similarly to $ATP$) used all over the organism.
Now, the body is not storing sugars in the form of glucose. Instead, they are stored as a glycogen (mainly in liver and muscles) which is a polysaccharide similar to starch. This is then quickly broken down to glucose as needed. But the body can keep only a small amount of glycogen (corresponding roughly to an hour of running, depending on one's fitness).
There is another form of storage of sugar. Body can convert it to fat. This is done when there is already enough glycogen in the organism. The body fat can then be reduced to acetyl-CoA (by lipolyses and then by beta oxidation), but this requires a lot more oxygen and so is not used when glycogen is at the disposal. But with regular exercise body can be trained to also burn greater proportion of fats than glycogen (this is of course necessary for long-range runners and cyclists because there is no way they would get enough energy just from glycogen).
To get a rough idea about the amount of mass you'll burn, read the calories content of some food. Sugar has something like 4kCal for 1g and fat 9kCal for 1g. One hour of running corresponds to something like 700kCal so if you are burning 50% sugars and 50% fat you'll be 100g lighter. All of these numbers are just very rough estimates depending on what kind of exercise you do and the general state of your body.

Note that you'll also lose lots of water and minerals during the exercise. But I am not counting this to the mass balance as you need to replenish those in order to be healthy. Also, gradually some muscles will form, so this will actually add weight.
A: When you exercise, you "burn" more glucose, the simplified reaction for which (from Wikipedia) is:
${\rm C_6H_{12}O_6 + 6~O_2 → 6~CO_2 + 6~H_2O}$
So when you exhale, the carbon in the carbon dioxide, and the hydrogen and the oxygen in the water vapor, came from the glucose being burned, thereby removing that mass from the body.
A: As others have pointed out, in metabolism you breathe in oxygen and exhale carbon dioxide with the net reaction being
$$CH_2O + O_2 \to CO_2 + H_2O$$
The carbon dioxide gets exhaled, and the water is lost through some combination of  things.  Let's think about whether this is a reasonable way to explain weight loss.
A breath is maybe one liter of air.  I breathe about ten times a minute, so that's 15,000 liters of air a day.  Air is about 20% oxygen, so 3,000 liters of oxygen.  Air weighs about 1kg/m^3, so about 3kg of oxygen.  The air you exhale still has some oxygen.  I don't know how good we are at extraction, but let's say we get a third of it, so 1kg of oxygen used by our bodies each day.
Carbon is lighter than oxygen and there's only one carbon exhaled per two oxygens, so that's about 300g of carbon lost per day.  Fat is basically a long carbon chain, so that says that if I don't eat, I'll lose about 300 g of body mass a day (assuming I do drink water).
It's not too bad a rough estimate to say that you lose weight through your nose.
I remember in middle school science class we learned about an experiment in which some guy a long time ago grew an entire tree in a pot, carefully covering the pot to keep stuff from getting into the soil.  The entire pot+tree system gained roughly the weight of the tree by the end.  So basically the tree got all its mass from the air.  You are pretty much an inverse tree.
A: It is like an automobile engine.  Think if you stopped putting gas in your car and just left it running.  It would use up the existing fuel and the output would be:
1) Energy to propel the car released from breaking the bonds in the gas 
2) Heat as a by product of the combustion and friction on the internal systems 
3) Leftovers from combustion that aren't used by the car (H2O, CO, CO2, etc)
Same principles at play in your body just a bit more biologically oriented.
A: Additionally, some fats (cholesterols, etc) are converted to bile and excreted by the liver into the digestive track.  On top of this many of the byproducts of cell death are excreted in this fashion.  That brown color to your poop is the product of red blood cells being broken down, and expelled by the liver.
Some minerals and other substances that are water soluble may be excreted by the kidneys.
A: When you burn up a log in a campfire, it weighs less than when you started, right?
Where did the mass go?
CO, CO2, H2O, and other combustion products.
It went up in smoke.
A candle? Same thing.
When you oxidize fat, sugar, whatever, what do you get? Same idea.
A: Here's a good way to think of it:
Consider your body like a leather wine bag... with a hole in the bottom.


*

*If you add less that what is leaking out, the bag will get smaller.

*If you add more than is leaking out, the bag will get larger.


As your body extracts the nutrients it requires out of stored cells (some carried in the blood stream and some stored as fat), those cells are disposed of (think poop).
With exercise, your body requires more fuel so it extracts nutrients out of more cells. If you are not putting enough in to replenish the supply, your body weight goes down.
A: It is carbon oxidation as mentioned before. When you breathe in $O_2$ and exhale $CO_2$ carbon is leaving your body and that is how you are losing weight.
What happens when you work out? You breathe more so more carbon mass can leave your body. Don't try to cheat and breathe hard while not exercizing or you will hyperventilate.
By the way, carbon oxidation is what also powers cars and campfires. Photosynthesis (using the sun to reduce $CO_2$ to $C$ and $O_2$) is the reverse of oxidation. Most organisms on earth are part of this sun cycle (you can trace all the food you and your car use back to plant), exceptions are some bacteria that can get energy from Lava and other means.
A: 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.
A: When you lose weight, it is mostly fat that disappears from under your skin (or someone else's skin). How does the fat get out? Most of it doesn't directly evaporate. Instead, it is burned much like fuel in your car. 
Fat contains mainly carbon, hydrogen, and oxygen, much like typical organic compounds. When it's "ideally" burned, only water and carbon dioxide - $H_2 O$ and $CO_2$ - are left. The gas $CO_2$ gets out by breathing and farting (not sure about the relative significance) - in which you only inhale $O_2$, the oxygen - while $H_2 O$ is released from the body by urination and sweating. Well, maybe, some other material may also be added to the excrements.
Of course, this is too idealized because by burning fats, one also produces some "less clean" compounds that are added both to urine as well as excrements (you asked, I am just answering). Farts also contain other gases that carry a certain small fraction of the mass of the burned fat - including methane. Biology is complicated but the difference between animals and cars is not so dramatic.
Gaining weight is the nearly reverse process except that the carbon is not taken from the air or $CO_2$ - we would have to be plants able to do photosynthesis do achieve this goal. Instead, it is taken from the food.
A: I recently lost a good deal of weight and would say it seems the water mostly leaves through urination, not exhalation.  Basically, I could tell when my weight was going to be decreasing because I'd have a lot of pee in the middle of the night, without having much to drink before bed.
As the other answers say respiration byproducts are co2 Whig is exhaled and water, which, I think, makes it's way to the bladder at some point. 
A: Weight loss occurs when  a large amount of calories or fat is turned into energy through excercise or activity.  That waste is then released from the body in one's urine, stool, & sweat.  When your body converts fat into accessible energy, the process generates heat that is used to regulate body temperature, according to MayoClinic.com. Once stored fat is converted to energy, your body uses it to fuel activity and metabolic functions in much the same way it uses immediate energy from food. MayoClinic.com notes that waste material produced during the conversion of body fat into energy, specifically water and carbon dioxide, leaves your body through urine, sweat and exhaling.
A: Years ago I worked for a weight loss supplement company. We had to reach out to the customers to help them along. Many of the men said while on the program it hurt to urinate. I was told that it is why it was so important to drink water In abundance esp during weight loss... BECAUSE when the fat breaks down it comes out through your urine. In a mans urethra, the opening at the tip of the penis is so small that if the fat loss is not diluted enough in the body it is too heavy to pass through that pin hole size opening in a man and becomes backed up and painful. When I lose weight I pee buckets very easily. And the more I drink water the more I lose the pounds on the scale even beyond the initial  5 lbs of water weight . But I've also been told that during rapid weight loss, your fecal excretion should appear greater than the volume of the food you've eaten. Particularly on the hCG diet I found this to be mostly true since the food volume is tiny compared to the " waste" going out. But I've also noticed that my mouth takes on a funky taste too, so maybe some of these posts have a point with the waste coming out in exhaling. Lastly , when losing weight there are times I feel so energetic that I can't even sleep earlier than 1:00 am. I just don't feel tired . But that could also be contributed to feeling lighter with less burden from the body weight... In summary - I think all explanations can be true based on my own observations during weight loss.
A: I encountered this question by accident and also had the same question a while ago. I found a TED talk about this subject with the title The mathematics of weight loss. The author is Ruben Meerman, he describes himself as: 

I am [sic!] surfer with a physics degree who fell in love with science
  communication.

The video of the talk can be found on the TED website (https://ed.ted.com/on/dgLmO0cP) or on YouTube.
As I understand the talk, the main weight loss is due to CO2 (breathing) and water. CO2 is the main effect though.
I attached three screenshots to show the look and feel.



A: Many answers have pointed out about the biochemical processes,  but have missed one important point.

The key of weight loss through exercise is not just about burning calories in your body,  but by consuming less calories than you actually burn. If you consume exactly the same calories that you burn, then there is very minimal weight loss.

So $$WtLoss_{Dieting} >> WtLoss_{Exercise}$$
A: Scientifically, the answer can be expressed in a very deep level of detail.  But ultimately, the answer is simple: Heat.
A: This is simple.  The answer is it leaves your body in the form of energy.
Your body burns sugars while producing things like heat (you get hot when exercising) and motion (you exercise).  So, we physically convert mass into energy which gets transfered mechanically into whatever we are doing.  That could be a distance traveled (walking/biking etc) or an increase in another objects potential energy (lifting weights etc)
