Lex and Alexei Luthor were their worlds' most problematic masterminds who had lengthy careers bedeviling their respective Metropolis, humanity and especially the Supermen.
As to the origins and legacies of these two tyrannical terrors, they were somewhat divergent. For Lex, it was when as a childhood friend of Superboy in Smallville of Earth-One, and experiment that backfired and cost him to lose his error created a rift between him and the Boy of Steel. As time went on, their adversarial relationship grew more and more hostile, until eventually one day he and Superman were drawn to a desert planet where Lex was viewed as a hero due to circumstances of their stay there. Renamed Lexor, this world became a safe haven for Lex as he would retreat there from time-to-time to lick his wounds from his various defeats at the hand of his arch-enemy. As that world's perceived sole superhero, Lex gained considerable acclaim with the inhabitants of this isolated globe in space, and eventually won the heart of Ardora. Together, they produced a son Lex Junior.
The background of Alexei had been clouded with ambiguity over time. When he first self-identified himself as a criminal genius, he was attempting to conquer the European nations of Galonia and Torsan. He was thwarted by the intervention of Earth-Two's Superman, which would culminate in a long standing grudge match between the two. However, his first appearance involved his directing remote controlled vehicles to cause mayhem on the roadways. This was his misguided attempted at seeking revenge for the death of his son.
Lex discovered that Lexor had ancient technology including a high-tech armor which he wore in an epic battle versus Superman high above the world. When a stray blast Luthor shot at Superman bounced off his foe and hit a stabilizing pole anchored to Lexor's planetary core, the world exploded. Losing his son and wife, Lex found a new level of hatred toward Superman.