Explain how pH changes can be minimised using a mixture of a weak acid and it's conjugate base

This mixture of a weak acid (HA) and it's conjugate base (A^-) acts as a buffer. A buffer is a solution with a highly stable pH. On addition of small amounts of either a strong acid or a strong base, this buffer will resist changes in pH.

Before addition:
The acid and it's conjugate base may react with each other, but this will result in no net change as both acid and base are reformed in the reaction. 

HA + A^- --> A^- + HA

Both the acid and it's conjugate base are weak so rarely react with the water present. 

As a result of both of these, the weak acid and it's conjugate base will remain in high concentration in the solution.

On addition of strong base:
As the base added is strong, it will react with the weak acid. The equation below demonstrates how the H⁺ from the weak acid is donated to the ^-OH group from the base. 

 ^-OH + HA --> A^- + H₂O

As the ^-OH group is used up by the reaction, the pH will only change minimally.

On addition of strong acid:
The H⁺ from the strong acid will react with the weak base to form the weak acid HA. 

H⁺ + A^-  --> HA

The additional H⁺ contributed by adding the strong acid is mopped up by the weak base. Therefore the pH will only change minimally