An absolute monarchy is, of course, a monarchy that performs within the bounds of absolutism. What does this mean, though? Well, it means that whoever is the monarch (usually a king or queen) of a given state has complete and total power. Anything that this monarch says in the imperative is taken as law, with no person able to contradict (except other states by force). Often, an absolute monarch will rule with the reasoning of a "divine right of kings" (throw in the suitable gender).
Most absolute monarchs had little regard for the peasants, who often lived in terrible conditions. The main goal of absolute monarchs with powerful states was usually to conquer more land, while those of weaker states usually tried to retain what land they ruled. War was often a part of the lives of absolute monarchs.
What many people consider to be the best example of an absolute monarch is King Louis XIV of France. Louis XIV expanded on the hunting lodge of his predecessors at Versailles into an immense palace. This served to consolidate his power (thus making it more absolute). Louis XIV would have many thousands of nobles at his palace at any given time, and they lived under very strict rules and customs (as specific as to designate who could sit when and on what chairs). This meticulous control of his palace served as an obvious example of the control Louis XIV exercised all over France, and is why he is considered the prime example of an absolute monarch.
Many, many countries have in the past had absolute monarchs, but in a (western*) trend starting with England during the reign of the Stuarts, European countries lessened the rule of monarchs until (in most cases) getting rid of them completely.
*If you have information on the time periods this occured elsewhere in the world, please /msg me.
Most information from lectures by professor Storch at the University of Wisconsin - Rock County. Additional information from Fourth Edition Western Civilization from 1300 by Jackson J. Spielvogel.