On of the definitions of acids and bases in acid/base reactions is that the acid is the species that donates a H⁺, while the base is the species that receives it.
As we can see, in the reaction the species HC₂H₃O₂ loses its H⁺ and pair up with Na⁺, so it is the acid, because it is donating the H⁺ in the reaction.
NaOH, on the other hand, is receiving the H⁺ and turning into H₂O, while its Na⁺ pairs with the other. This means that NaOH is the base, since it is receiving the H⁺.
The conjugate acid is the acid that is formed in the reaction by the species that received the H⁺, that is, the base reactant receives the H⁺ and turn itself into its conjugate acid.
The conjugate base, on the other hand, is the species formed by the acid after it donated the H⁺.
The acid, HC₂H₃O₂, turned into NaC₂H₃O₂, so this is the conjugate base.
The base, NaOH, turned into H₂O, so this is the conjugate acid.
Thus, we have:
Acid: HC₂H₃O₂
Base: NaOH
Conjugate Acid: H₂O
Conjugate Base: NaC₂H₃O₂