As a concept, enthalpy ties in closely with both the first and the second laws of thermodynamics. You won't find much mention of it in physics textbooks, however, because its use is predominantly biochemical.
Enthalpy is the change in heat following a reaction taking place at constant pressure. If heat is given off by a reaction, enthalpy is a positive number; if heat is taken in instead, enthalpy is necessarily a negative value.
Why "at a constant pressure"? Well, in the normal cellular environment, reactions approximate constant pressure conditions. Therefore, it is this state that we wish to consider when calculating heat changes. If the environment is one of (say) constant volume, the heat change would have to be calculated differently.
Enthalpy (ΔH) is defined by the following equation:
ΔH = ΔU + PΔV
(recall that U is the internal energy of a system; P is pressure, and V is volume)
This equation looks difficult to remember, but its really easy to derive from the equation of the first law of thermodynamics. Watch:
ΔU = Q - W
∴ Q = ΔU + W
And, since (from physics) W = PΔV:
Q = ΔU + PΔV
∴ (since Q at constant pressure = ΔH):
ΔH = ΔU + PΔV
Personally, I find the above derivation helpful - it reminds me that the enthalpy equation isn't really a new equation, and it allows me to work it out if I forget it!
Finally, here's why enthalpy is useful to biologists.
One of the most fundamental questions facing a biochemist is whether or not a reaction will occur by itself ('spontaneously', although the word doesn't imply anything about the speed of the reaction), or whether the reverse reaction will do so. For instance, glycogen is a polymer consisting of multiple glucose molecules. If we put glucose and glycogen into a cell's water, will the glycogen tend to break down into glucose, or will the glucose start assembling into glycogen? This is an important question, because if the cell 'wants' a non-spontaneous, unfavourable reaction to occur (in this case, transforming glucose molecules into glycogen), it''ll need to expend energy to accomplish this. On the other hand, if it 'wants' a favourable (spontaneous) reaction to occur, it can just sit back and wait (although in practice, it would still want to use enzymes to dramatically increase the reaction's rate). Which way a particular reaction will go is determined by two main clusters of variables. And one of these is... enthalpy, of course.
Reactions that have a positive value for ΔH are called exothermic reactions, and give off heat (by definition). Reactions that have a negative value for ΔH are called endothermic reactions, and require heat to be added in order for them to proceed. Spontaneous reactions tend to be exothermic although, CRUCIALLY, not all spontaneous reactions are so; there is still the other variable cluster to consider (we'll get to that a following post).
Take home message: Enthalpy is a measure of the heat given off or taken in by a reaction occurring at a constant pressure (i.e. under biological conditions). Exothermic reactions give off heat (ΔH > 0), whereas endothermic reactions require heat to be added. Enthalpy is one of two sets of variables to consider in determining whether or not a reaction will happen spontaneously or not.
The last two paragraphs contained, until now, a rather careless error. Originally, I had said that reactions that gave off heat were "exergonic", and that reactions that required heat were "endergonic". This is false - the correct terms are "exothermic" and "endothermic" respectively. "Exergonic" and "endergonic" refer to the overall changes in free energy - the former implying that the reaction is thermodynamically favoured, with the latter implying the reverse. More on this in subsequent posts, but I thought it'd be useful to point out the error to stop it being repeated!