There is a common fallacy that their name means 'breastless' because they cut off their right breasts to make archery easier. This, however, is a false definition. Their name may come from Armenian maza, 'moon'.
They were a race of warrior women said to live in what is now Turkey in mythological times, that is, when Greece was going through its prehistoric Minoan and Mycenaean ages. The story goes that Theseus, King of Athens, abducted their queen and the Amazons, allied with the Scythians, invaded Greece to retrieve her.
Did such an invasion ever take place? It's possible, because the Amazons in reality seem to be based on female Scythian women warriors, graves of whom have been discovered. If a Scythian army attacked Athens in prehistoric times, later legend may have separated the attackers into two races, Scythians and Amazons.
The Amazons were also said to have taken part in the Trojan War where their queen was slain by Achilles. Later the term was applied to warrior women said to be found in South America. This caused the River Marañon to have its name changed to the Amazon.