The White Lion (who was known as the Guardian) was the mysterious guardian spirit of the mystical realm of Oriande. It stands watch over the entrance - suitably located within a white hole - ready to attack anybody who dares to approach, except for those with ‘The Mark of the Chosen’. This is presumably a genetic trait passed down among Alteans that causes their markings to glow and permits them access to Oriande.
In the episode "The White Lion", Allura and Lotor seek to enter Oriande to uncover the wisdom of ancient Altean alchemy. First the Castle of Lions, then Voltron attempt to enter the white hole, but are attacked by the looming, spectral head of the White Lion. Then, the Team notices that Allura's Altean markings are glowing, and that glowing purple marks have also appeared on Lotor's cheeks. Taking this to be a sign of permittance, the two 'Chosen' venture out into the white hole and are granted entry into Oriande.
Inside, they find a heavenly pink land of crumbling marble structures. There, they encounter the real White Lion. He fights them separately, testing their willingness to kill and gauging their purity of spirit. Allura, who kneels and spreads her arms in submission, is deemed worthy. Lotor, who draws his sword and tries to kill the White Lion, is deemed unworthy. Allura transcends Oriande and learns the secrets of Quintessence manipulation.
Additionally, the White Lion appears to have been the inspiration for the Voltron Lions, since King Alfor is believed to have ventured in search of Oriande and found the white hole.