The Bronsted-Lowry acid/base definition defines acids as proton donors and bases as proton acceptors. A conjugate acid/base refers to the compound formed after an acid-base reaction. For example, in the generic reaction:
where HA is the acid, B- is the base, HB is the protoned base, and A- is the acid after the proton transfer. Since the acid (HA) released a proton, it became A-, which is basic since it can now accept a proton. This means that A- is HA's conjugate base. Similarly, since the base (B-) accepted a proton to form HB, HB is now an acidic since it has a proton to release. This means that HB is B-'s conjugate acid.