A τύραννος (túrannos) was simply an "usurper", someone who had seized power through unconstitutional or extralegal methods outside of normal rules of succession and inheritance that normal monarchs followed. It didn't quite have the same pejorative connotation that exists today.

A Tyrant was a ruler who gained absolute power through non-constitutional, generally violent ways. Herodotus recalled the story of one such Tyrant, Pisistratus of Athens and his unusual venture into acquiring sovereignty over the Athenian people. He is described as the ideal Tyrant, one who does not abuse his position and works to genuinely improve the lot of his people. However no matter how benign or even benevolent his rule Pisistratus is still described as having the Athenians "held down," showing a relatively negative outlook on Tyranny.

Conversely, monarchies were established through hereditary succesion and were able to be somewhat limited in policymaking by assemblies of citizens and constitutional law. Sparta's dual monarchy in particular was described in detail by Xenophon in his Lakedaimonion Politeia as consisting of the kings, whom "everyone rises for" except for the ephoroi (the citizen assembly) "who remain seating." Furthermore the kings and the assembly both were to take a monthly oath in which the kings swore to uphold the nómoi (law) of land, while the assembly swore on behalf of the polis to remain faithful to a law abiding monarchy. Finally he describes the ideal Greek monarchy as one in which the Kings did not develop the "arrogance of Tyrants" and likewise the assembly did not "grow envious" of the kings power.

Bearing the other two answers in mind, were there any Tyrants who founded monarchies? By that I mean, they were regarded by some as a tyrant in their time but their line persisted and history looks back on the "Kingdom of X" rather than the "Republic of X, interrupted by this one guy".