So Hippolytus is named after his mother, who judging by his attitudes towards women he probably doesn’t remember that well (though watching your mother try to kill your father and stepmother can polarize a child’s relationship with his mother pretty severely. Read more about that on the Theseus page.)
The reason I say he couldn’t have really known her that well is that the Amazons would not let him get away with that whole “smart women are dangerous” stuff. In the Iliad, Homer calls the Amazons the Antianeirai, “those who fight like men.” And that’s pretty much what they were known for: being a tribe of warrior women who fought and functioned as men did in other Hellenistic societies.
Women were warriors, merchants, religious and political leaders, physicians, etc. They held every position an Athenian woman in Euripides’ time would have been barred from participating in. Some accounts have them keeping men as slaves, but many also say that they would mate with a neighboring tribe once a year in order to keep the society from dying out. The whole cutting off one breast to improve archery thing is pretty suspicious considering a lot of them would end up raising children later, but mutilation of the body is a ceremonial part of many societies so it’s entirely possible.
The homeland of the Amazons moves about quite a bit in Classical sources, from Asia Minor to the Ukraine and Crimea. Most Classical historians can’t agree on exactly where they came from, but they all concur that they enjoy invading people. A lot. There are accounts of Amazon invasions of Lycia (now part of modern day Turkey), Libya (Africa), Phrygia (Turkey again), and finally, Athens itself in what is usually referred to as the Attican War.
This motif of Greek heroes (namely a young Theseus and older Heracles) fighting the Amazons makes up a whole genre of Greek art, called Amazonomachy (“battle with Amazons.”) Turns out even in the Bronze and Classical periods, men liked watching women kick ass.
Hippolyta (or Hippolyte) was the queen of the Amazons during the majority of the Age of Heroes. (That’s when Perseus, Heracles, and Theseus went about their business.) Heracles was told to capture her girdle as a part of his twelve labors. (For a hilarious and foulmouthed retelling of the labors, go here.) So he and his little buddy Theseus, fresh from abandoning Ariadne and killing the Minotaur head off to wherever the Amazons live at the time.
And somehow (there are too many variations on the story to pick one) Heracles gets the girdle. Maybe he kills with her. Maybe he sleeps with her. Maybe he does both. Anyhoo, Heracles got what he came for, and Theseus is like “I want something too.” so he picks up a chick, either Hippolyta’s sister, or the queen herself, and he and Heracles leave. And that upsets the Amazons a lot, so they fight them, and they lose.
So the Amazons bide their time and reappear again after Theseus has decided to settle down and marry a Greek princess (Phaedra) rather than have a common law marriage with a barbarian Amazon. (Because breaking up common law marriages with barbarians end so well, ask Jason of the Argonauts.) The Amazons tried to invade Attica and were repelled again, and either Hippolyta died or she was already dead and her sister (Hippolytus’ mother) died. And everyone disagrees about which side Theseus’ Amazon girlfriend was on. Some say she led the Amazons, others that she fought at Theseus’ side. Either way, she’s dead and Hippolytus ends up a bastard and basically an orphan because he’s got barbarian blood and his parents weren’t really married. (Again, read Euripides’ Medea for all the fun xenophobic reasons why barbarian women weren’t real wives.)
Despite this little setback, the Amazons continued to appear in “history” even to the time of Alexander the Great, when the then Amazon Queen was rumored to have conceived a child with him.
These women, whether real, or an elaborate male fantasy meant to demonize female power, persisted as a motif in myth and “history” for centuries. Hippolyta and Theseus even make an idealized appearance in Shakespeare’s Midsummer Night’s Dream, which is set on the eve of their wedding. In the opening scene, Theseus tells Hippolyta
Hippolyta, I woo’d thee with my sword,
And won thy love, doing thee injuries;
But I will wed thee in another key,
With pomp, with triumph and with revelling.