There are several ways household ammonia and bleach can react. All of them are dangerous.
Reaction type 1: Ammonia directly reacts with bleach to form hydrazine (N2H4), which, in addition to being extremely poisonous, can burn even in the absence of air! It explodes on contact with rust!
2NH3 + NaOCl -----> N2H4 + NaCl + H2O
Reaction type 2: Bleach hydrolyzes into sodium hydroxide and hypochlorous acid, which in turn decompose into chlorine gas and nascent oxygen (both poisonous). The chlorine gas in turn reacts with the ammonia to form chloramines, also very poisonous.
NaOCl -----> NaOH + HOCl
HOCl ---> HCl + O (monatomic oxygen)
NaOCl + 2HCl -----> Cl2 + NaCl + H2O
2NH3 + Cl2 -------> 2NH2Cl (chloramine)
4NH3 + 2Cl2 ------> 2NHCl2 (dichloramine)
6NH3 + 3Cl2 ------> NCl3 (trichloramine or nitrogen trichloride)