Ammonium carbonate is analogous to the other bicarbonate and carbonate salts you see - baking soda. In the presence of acid; or just sufficient heat, it will offgas ammonia and carbon dioxide, hopefully leaving pleasing bubbles in whatever you're making.
Whatever you're making better not have much moisture, though. Ammonia loves water, and any residual water will hold onto enough to give whatever food you're making a profound stench. This pretty much limits you to crackers; ammonium carbonate-leavened banana bread will not be pleasant.
The salt has been around for a long time but has been supplanted almost entirely by baking soda (sodium bicarbonate, NaHCO3), which adds a little sodium to the recipe but has the distinct advantage of not imparting the taste of kitchen cleaner to your cupcakes.
The strange thing about the smell of ammonia is that on its own it is quite pleasant in small doses (at least in my opinion) but when it is combined with anything else it smells foul.
For instance if you are precipitating protein from fish guts with ammonium sulfate it smells horribly rancid even though the fish is fresh and not bad smelling when you begin.
I maintain that thiols smell worse...
but that's a whole different stinky issue...
I thought "ammonium carbonate" was actually ammonium carbamate, NH2COO- NH4+. When a strong primary amine, such as ethyl 3-aminopropionate reacts with air it forms RNHCOO-NH4-R and not (RNH3)2 CO3. The nmr spectrum clearly shows that the salt is a carbamate salt and not a carbonate.