What happens when a base is added to a system with a buffer and which part of the buffer reacts with the added base?

A buffer consists of both an acid and a base. So if a base is added the acid of the buffer reacts with the added base to limit the pH. If an acid is added the base of the buffer reacts with the added acid to limit the pH change.

Suppose we have a buffer with acetic acid (HAc) and sodium acetate (NaAc).
So if we add HCl to the buffer the reaction is with NaAc to form the weak acid HAc and that limits the pH change.
Ac^- + HCl ==> HAc + Cl^-
If we add NaOH to the buffer the reaction is with HAc to form H2O and the small change in HAc and water limits the pH change.