Product of the reaction is NH3(aq). It is sometimes written as NH4OH (there is plenty of water around), but from what I remember existence of this compound is rather dubious.
Ammonia is pretty well soluble in water, but yes, such solutions - if left open - will slowly loose the ammonia.
As presence of NH4OH is dubious, there is not much sense in discussing its reactions. Note that it is possible we in fact deal with the multistep equilibrium
NH3 + H2O <-> NH4OH <-> NH4+ + OH-