In the original Greek meaning, a tyrant was anyone who overturned the established government of a city-state, usually through the use of popular support, to establish himself as dictator, or the heir of such a person. The term now implies a cruel persecutor who treads on the welfare of his people, because some tyrants did indeed have these characteristics. It has also been used by extension of non-governmental figures, such as patriarchs and bullies. The title was first given to Pisistratus of Athens in 560 BC.
The heyday of the tyrants was the early 6th century BC. During this time, many governments in the Aegean world were overthrown. It was during this time that Persia first made inroads into Greece, as many tyrants sought Persian help against forces seeking to remove them.
Tyrants were generally installed by popular coups, and were often popular rulers, at least in the early part of their reigns. For instance, Pisistratus was remembered for an episode (related by Aristotle but possibly fictional) in which he exempted a farmer from taxation because of the particular barrenness of his plot. Pisistratus' sons Hippias and Hipparchus, on the other hand, were overthrown, and Hipparchus was assassinated.
Later ancient Greeks, as well as the Roman Republicans, were generally quite wary of anyone seeking to implement a popular coup.