What is the difference between ammonium and ammonia?

Ammonia and ammonium differ by one hydrogen ion (H+). Ammonia is the common name for the compound nitrogen trihydride, which has the formula NH_3 . Ammonium is the name of the ion NH_4^+ . An ion is an atom or molecule that has a net charge. 
Ammonia is a base, meaning it produces OH^- in solution. When ammonia reacts with water, an H+ ion is transferred from water to ammonia, forming ammonium ion, according to this equation:
NH_3 + H_2O -> NH_4^+ + OH^- 
Ammonia and ammonium ion are a conjugate acid-base pair. Ammonium ion is the conjugate acid of ammonia because it forms when when the base ammonia (NH_3) accepts a proton (H+ ion), as seen in the equation above. When the NH_4^+ ion donates a proton, its conjugate base NH_3 is formed. This occurs in the reverse reaction:
NH_4^+ + OH^(-) -> NH_3 + H_2O