The Henderson-Hasselbalch equation is a statement describing the relationship between the concentration of an acid in solution and the pH of the solution. It is one of the fundamental principles of chemistry. It is particularly useful in biochemistry where it provides a general solution to the quantitative treatment of acid-base equilibrium in biological systems. The equation was developed by a large group of scientists in the early 20th century and is named after two of the more central figures in the study.
Let's say we have a weak acid HA of which we know the acid dissociation constant Ka (which can easily be found in chemical literature; there are literally books full of acid dissociation constants). The basic chemistry of acids states that:
HA H+ + A-
Basically, an acid exists both as HA and its respective ions. We also know what the acid dissociation constant represents for this acid:
[H+] * [A-]
Ka = -----------
[HA]
[H+] means the exact amount of H+ in a specific amount of solution, virtually always in moles per liter. We then rearrange these equations into something more useful:
Ka * [HA]
[H+] = ---------
[A-]
The definition of pH is log [H+], where by log we are referring to log base 10. So, we take the log base 10 of both sides:
Ka * [HA]
log [H+] = log = ---------
[A-]
Which, if we translate into a pH statement and juggle, reveals the Henderson-Hasselbalch equation:
[A-]
pH = log Ka - log ----
[HA]
The equation basically gives a quick way of figuring out the pH of a solution if you know what acid you are starting with, the concentration of the acid, and the concentration of its conjugate base. This makes the creation of solutions with specific pH's much simpler, aids greatly in the creation of buffer solutions, and also provides a very useful tool in the basic analysis of many complex chemical systems. | 
