The Puranic myth most responsible for fixing the asuras as the devas' standing rivals is the churning of the ocean of milk, samudra manthana, told at length in the Bhagavata Purana and the Vishnu Purana. Weakened by a sage's curse, the devas can no longer hold their own against the asuras and, on Vishnu's advice, agree to a temporary alliance: the two sides together use the serpent Vasuki as a churning rope wound around Mount Mandara to churn the cosmic ocean and bring up amrita, the nectar of immortality, along with a series of other treasures and, notably, poison that Shiva swallows to save creation from it.
When the amrita finally emerges, the alliance collapses immediately, the asuras seize it first, and Vishnu, in the form of the enchantress Mohini, tricks them into surrendering it back so the devas can drink instead, cementing an enmity the story treats as permanent from that point on. The episode is the clearest statement in the tradition of what the asura-deva rivalry is actually about: not abstract good against evil, but two related families of powerful beings contesting a finite, extremely valuable resource that only one side can ultimately keep.