It's not uncommon to hear people say things like, "Carbon neutral? What's that? Wouldn't you have to stop breathing? <huh huh> <huh huh>"
The carbon in the carbon dioxide that you exhale is a product of cellular respiration inside your cells. It comes from breaking down sugar (a carbohydrate). The process liberates the carbon from the sugar so that it can bond with the oxygen that you inhale to form carbon dioxide and water (e.g., urine or "pee"), and this releases energy since carbon would much rather be a part of carbon dioxide than sugar. This energy is what you use to go about your day and say statements like the above.
So where does the sugar come from? It what is produced in the photosynthesis that goes on in plants. Photosynthesis uses energy from the sun to reshuffle water and carbon dioxide into oxygen and sugar.
That's right. The carbon you exhale comes from sugar that could not have been made without pulling carbon out of the air.
So that's why breathing is carbon neutral.
This is also why combustion, which is related to cellular respiration, of pure biofuels (hydrocarbons) is POTENTIALLY carbon neutral (see notes about sustainable yeilds); the carbon in those fuels was originally pulled out of the air by a plant somewhere, and hopefully more plants will pull it out of the air after combustion. Of course, there are lots of other things involved in combustion (e.g., processing and transporation of biofuels as well as the consumption that goes on in the process of burning those fuels) that are not carbon neutral (and, again, there's that issue of burning so many plants that you don't have any left to take the carbon out of the air; again, see sustainable yield).
You should consider these things when you think about livestock as well. Breathing does not prevent something from being carbon neutral.
So what isn't carbon neutral? If you are releasing carbon that did not (recently) come from a plant source, you are probably not being carbon neutral. Burning fossil fuels is liberating carbon that has been buried for (hundreds of) millions of years. This is why the word fossil is used. This carbon would have stayed buried if it was not for human intervention. This causes a stability problem on the surface of the earth. The carbon cycle gets out of whack because that extra carbon does not go back to its source. In fact, the only way to get that carbon back to where it came from is to have a mass extinction and then millions of years of geological processes.
So that's the scoop on carbon neutrality.