The Mycenaeans lived during the late Bronze Age in Greece, from about 1600 to 1200 B.C. They are called Mycenaeans after their capital at Mycenae. The period in which they lived (Mycenaean), in ceramic terms is known as the Late Helladic period.
We don't know what happened to the Minoans in Crete, but they seem to have been taken over by the Mycenaeans from the Greek mainland, because after 1500 a new writing dominated Cretan records, Linear B, the language of the Mycenaeans.
We know little about the Mycenaeans except from archaeological remains and Homer's epics, in which the Mycenaeans fought against Troy. Among other areas of ignorance, we do not know whether it was the Dorian invasion, natural catastrophes, or something else that ended the Mycenaean civilization, but whatever it was, the end of the Mycenaeans ushered in a "Dark Age" in Greece.
For issues of the chronology, see "Review of Aegean Prehistory VI: The Palatial Bronze Age of the Southern and Central Greek Mainland," by Cynthia W. Shelmerdine. American Journal of Archaeology, Vol. 101, No. 3. (Jul., 1997), pp. 537-585.
In Greek legend, the hero Perseus founded Mycenae.