Let us take the example of Bicarbonate being placed in water to create Carbonic Acid and Hydronium ions.
HCO_3^(-) + H_2O -> H_2CO_3 + OH^-HCO−3+H2O→H2CO3+OH− Base + Acid -> Conj A + Conj BBase+Acid→ConjA+ConjB
On the reactant side of the equation the water (H_2OH2O) is the Acid as it is the hydrogen ion donor to the bicarbonate (HCO_3^-HCO−3) which is the Base as it receives the hydrogen ion..
On the product side the Carbonic Acid (H_2CO_3H2CO3) is the Conjugate Acid as it is the hydrogen donor to the Conjugate Base (OH^-OH−) as it receives the hydrogen ion.
HCO_3^(-) -> H_2CO_3 HCO−3→H2CO3 Base -> Conj ABase→ConjA
The Carbonic Acid (H_2CO_3H2CO3) is the Conjugate Acid pair to the Base bicarbonate (HCO_3^-HCO−3) .
H_2O -> OH^-H2O→OH− Acid -> Conj BAcid→ConjB
The Hydroxide (OH^-OH−) is the Conjugate Base pair to the Water (H_2O#).