Ammonia is a chemical by-product of tobacco fermentation. During the curing process, tobacco leaves are stacked into pilones (large piles in the curing barn) that heat up under their own weight; the controlled fermentation breaks down sugars, removes bitter compounds, and as a side effect produces ammonia. Manufacturers ferment carefully and at length precisely to bleed off as much of it as possible before the leaves are rolled. Some always remains.
After rolling, the cigar continues to off-gas slowly for months. The shape of that curve depends on the cigar, but by most accounts the sharp ammoniac smell fades substantially within the first few months, is mostly gone by the end of the first year, and is essentially gone by the second.
You can smell it on a fresh cigar if you hold it an inch from your face and breathe in slowly. A clean, sweet, woody, leathery, or barnyard nose is a cigar where the ammonia has cleared. A sharp note that pulls your nostrils back is a cigar still off-gassing.
If you smell ammonia in your humidor itself (not just on individual cigars), it usually means a fresh box was added recently and the off-gas is filling the enclosed space. Crack the lid for an hour, redistribute the cigars, and the smell typically resolves.
The longer treatment of ammonia and the off-gassing curve sits in How to Rest a Cigar.