The origins of some constellation myths are heavily debated. Gemini mythology, luckily, is pretty straightforward.
By all accounts, the twins represented in Gemini mythology are Castor and Pollux of Greek mythology. They share the same mother, Leda (Greek mythology), but have different fathers. Castor's father is Tyndarus, the King of Sparta, and Leda's husband. Pollux's father is the god Zeus (Greek mythology). As such, Pollux is an immortal while his twin brother Castor is mortal.
The twins were young, handsome, and adventurous. They took part in many adventures together and were well known for their livelihood and curiosity. Castor was a renowned horseman, while Pollux was known for his great strength.
Their sister is the beautiful Helen of Troy, whom the great Trojan War is fought over. The twins not only took part in that war together, but were also Argonauts on the quest for the Golden Fleece.
Where Gemini mythology comes into play is when Castor, being mortal, finally dies. Having spent their whole lives together, Pollux is distraught. He doesn't want to live without his twin brother, but since he is immortal, there is nothing he can do. He begs his father, Zeus, for help.
Zeus decides that rather than killing Pollux so he can be with Castor, he makes Castor immortal also, and the two of them get to live together forever as the constellation Gemini.
Perhaps the reason that this story is rarely contested is because two of the actual stars in the constellation of Gemini are named "Castor" and "Pollux". This is a rare case where astronomy and mythology actually agree, and thus, Gemini mythology is born.