Acid + Base ---> Salt + Water
is a useful general rule, but don't get too attached to it, as it isn't that useful when you are using Lowry-Bronsted definitions of acids and bases. In the case of ammonium nitrate formation, you can look at the reaction as a Lowry-Bronsted acid-base reaction, with the transfer of a proton from the acid (HNO<sub>3</sub>) to the base (NH<sub>3</sub>), forming (respectively) their conjugate base and conjugate acid. ie:
HNO<sub>3 (aq)</sub> + NH<sub>3 (aq)</sub> ---> NO<sub>3</sub><sup>-</sup><sub> (aq)</sub> + NH<sub>4</sub><sup>+</sup><sub> (aq)</sub>
This is an example of the general reaction:
acid_1 + base_1 ---> conjugate base of acid_1 + conjugate acid of base_1