Some Greeks from Euboea established a settlement (Aenaria or Pithecusae) in the Bay of Naples around 770 B.C. (The distance from Rome to Naples is 117.49 m. or 189.07 k. to the southeast.) Excavations there show iron-working, which supports a belief that the Greeks went to Italy in pursuit of metals. The areas settled by the Greeks may have been colonies or trading posts or both.
Later Greeks moved to the western Mediterranean in search of a better life. Shortly after the settlement of Pithecusae, there was a colony at Cumae, which was followed by other colonies in southern Italy and Sicily.
The colonists did well and so one of the colonies, Sybaris, became synonymous with luxury (sybarite).
The name Magna Graecia was in use to apply to southern Italy by the 5th century. To the Greeks, the area was known as Megale Hellas [see map of southern Italy].
Source (and for more information): T.J. Cornell The Beginnings of Rome
People may use the term Magna Graecia in two slightly different ways. Either it includes the Greek islands or refers strictly to the mainland Greek-settled territories of southern Italy, according to "Chapter 18 - Early Rome and Italy," in The Cambridge Economic History of the Greco-Roman World, edited by Walter Scheidel, Ian Morris, Richard P. Saller.