For example, how could you make a buffer with equal concentrations of acid and conjugate base? If you add more conjugate base, it will shift the equilibrium to form more acid, which will keep the proportions of each the same. So how can you achieve equal concentrations if this is happening?
The Henderson Hasselbalc equation is a good way to calculate both the pH and the concentration of acid/conjugate base in an equilibrium solution. According to it:
pH = pKa + log([A-]/[HA])
Since you want to know when [A-] = [HA], you are looking for when the pH is equal to the pKa. This means that you will have equal concentrations of the acid/conjugate base whenever your pH is equal to your pKa, which you can find by monitoring the pH. While adding more conjugate base will shift the equilibrium towards acid formation, it will normally increase the acid concentration less than the amount of base that you added.