A "solution" is a mix of any two substances; buffers are general "aqueous solutions," which means they dissolve in water. "Bases" are substances that accept or donate electrons in compounds, and "acids" are substances that easily react with bases; thus, a buffer is a mixture that does not react easily because it is strong in its acid or base already. "pH" is a measure of how acidic or basic a solution is, with pure water being in the middle of the scale at 7; substances less than 7 are acidic and substances greater than 7 are basic.
The equation for determining the pH of a a weak acid buffer solution is pH = pKa + log[salt]/[base], where pKa is the acid dissociation constant and [salt] and [base] are the molar concentrations of those two substances in the solution. The equation for determining the pH of a weak basic buffer solution is the same, but instead find the log of the molar concentration of the base divided by the molar concentration of the salt (effectively, the flip of the weak salt equation).
In addition to determining the pH of a buffer, it is also useful to determine the buffer capacity, which reveals how resistant the buffer is to changes. The equation for buffer capacity is to [delta B/delta pH], which means you add a certain amount of an acid or a base to the substance (delta B) and divide it by the change in the substance's pH (delta pH). Delta B should be noted in grams, so the final expression would be grams per pH.
Buffers are an essential part of biology because they ensure stability at the cellular levels. For example, in the human body, blood requires a pH level of around 7.4 in order to produce enzymes, and thus the hearts, lungs and kidneys work together in order to ensure that the pH level does not rise above or below these levels. The primary buffer in the human body is carbonic-acid-bicarbonate.