I was under the understanding that tyrants were defined by their lack of lawful authority to rule and being usurpers, but a video we watched in class suggests that many city-states built into their constitutions the ability to elect a tyrant if they were unhappy with the current rulers. I'd really appreciate a good explanation of what precisely a tyrant was!
I can't speak for other parts of Greece but I can say something about ancient Athens.
The title "tyrants" began after Peisistratus (~6th century BC) assumed power after Solon, a reformer whom Athenians elected, went on a ten year holiday. "Tyrant" didn't have its modern definition meaning an oppressive dictator. It just meant that the leadership was usurped and had unlimited power (autocratic) because tyrants had control of a very strong military force. Peisistratus technically wasn't allowed to take leadership of Athens because Solon demanded that Athenians uphold his reforms before he comes back. The Athenians couldn't stop Peisistratus from assuming power because he had the strongest army so Athenians couldn't do much about that but to give in. However, Peisistratus wasn't some mad, evil dictator. Despite his authoritarian power, he made Athens incredibly prosperous economically. His rule also ended factionalism which was partly caused by Solon's reforms.
After Peisistratus, his son Hippias came to power. Hippias's brother Hipparchus was assassinated by Harmodios and Aristogeiton. The plot also eventually killed Harmodios and Aristogeiton. After the assassination, Hippias became incredibly paranoid and cruel and ruled Athens with an iron fist.
Angered and dissastified by Hippias's reign and the Peisistratids' power over Athens, the Alcmaeonidae family of Athens (who Peisistratus exiled during his rule) called upon Spartans to help them overthrow Hippias. Cleomenes I, the Spartan king, would successfully help the Alcmaeonids take over the Acropolis and force the Peisistratids (Hippias and his family) to go into exile. Hippias fled to Persia and would establish his influence there which would come up later during the First Persian War at the Battle of Marathon. After the Peisistratids were out, Athens began its path toward democracy.
This is a brief summary of where Athenian tyrants came from. During the periods before Solon, Athens was ruled by a series of kings and aristocracies. Although they had a lot of power over the demos (Athenian commoner), they didn't hold the title "tyrant." As for where the term comes from, ancient thinkers like Plato and Aristotle had written about tyrants in their history. Aristotle's Athenian Constitution and Plutarch's "Life of Solon" are my sources and they talk extensively about the change of political power in Athens. Herodotus and Thucydides had also written about this period.