Bronsted Lowry Theory of Acids and Bases
An acid is a substance that donates H+. A base is a substance that accepts a proton(H+)
H3O+ + Cl- à HCl + H2O
H3O+ is simply water with acid in it. The proton will be strongly attracted to water, and form an H3O+ ion. In formulas such as pH = -log [H+], you can interchange H+ with H3O+. Whether this ion actually exists has been the subject of recent research.
In any case, since H3O+ becomes water, it donates or loses an H+. This makes H3O+ an acid. Less obvious is that relative to H3O+, Cl- is acting as a base because in becoming HCl it accepts a proton.
NH3 + HCO3- = NH4+ + CO3-2
In the forward reaction, NH3, in becoming NH4+, is accepting a proton, so NH3 is acting as a base. HCO3- is losing a proton (compare to what it became: CO3-2), so in this case, HCO3- is acting as a Bronsted acid.
For the reverse reaction, NH4+ is the acid, and CO3-2, the base. The adjective conjugate is often attached to the acid and base for the reverse reaction.