Reactions are classified as exergonic or endergonic by the change in a quantity called "Gibbs free energy." Unlike endergonic reactions, an exergonic reaction can occur spontaneously, without a need to input work. That doesn't mean a reaction will necessarily occur simply because it's exergonic -- the rate at which the reaction occurs could be so slow that it will never happen on a timescale you care about.
Gibbs Free Energy
Gibbs free energy is not called "free energy" because there's no price tag, but because it measures how much non-mechanical work a system can do. If the reactants in a process have higher Gibbs free energy than the products, the process is called exergonic, meaning that it releases energy. Another way to say this is to describe the reaction as thermodynamically spontaneous, meaning you don't have to do work to make the reaction happen.
Exothermic vs. Exergonic
Many, but not all, exergonic reactions are exothermic, which means they release heat. A reaction can actually be exergonic, however, and yet absorb heat, or be endothermic. Consequently, exothermic and exergonic do not necessarily go together. The key difference between them lies in the difference between work versus heat; an exergonic process releases energy through work, whereas an exothermic process releases energy through heat. Moreover, a process may be exergonic at some temperatures but not at others.
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Entropy vs. Enthalpy
Nineteenth-century chemists found spontaneous endothermic reactions quite puzzling; they reasoned that a reaction should be spontaneous if it releases heat. What they were missing was the role of entropy, which is a measure of the amount of energy unavailable for work in a system. If we consider the system as well as its surroundings, a process will be exergonic if it causes a net increase in entropy. Releasing heat to the surroundings causes the entropy to increase, but such a reaction can still absorb heat and be exergonic if the entropy of the system increases by an even larger amount.
Evaporation -- the process whereby a liquid turns into a gas -- is associated with a very large positive change in entropy. Exergonic reactions that absorb heat are often reactions that release a gas as one of the products. As the temperature increases, these reactions will become more exergonic. An exothermic reaction that releases heat, by contrast, will be more exergonic at lower temperatures than at higher ones. All of these considerations play a role in determining whether a reaction will be spontaneous.