A father's boon, and its blind spot
Hiranyakashipu earned a boon from Brahma after long austerity, making him nearly impossible to kill: not by man or animal, not indoors or outdoors, not on the ground or in the sky, not by day or by night, and not by any weapon. Believing himself effectively immortal, he declared that he alone should be worshipped in place of the gods — and forbade the name of Vishnu from being spoken in his kingdom.
A son who wouldn't obey
His own son Prahlada, however, was devoted to Vishnu from birth and refused to stop worshipping him, even as a young child. Hiranyakashipu tried repeatedly to kill his son — poison, wild elephants, being thrown from a cliff, being placed in fire while sitting in his aunt Holika's lap (she alone was fire-proof, but the boon failed her because she misused it, and Prahlada walked out unharmed). Nothing worked; each time, Prahlada's devotion protected him.
The pillar, and Narasimha
Finally, enraged, Hiranyakashipu demanded to know if Vishnu was truly everywhere, as Prahlada claimed — even inside a palace pillar. When he struck the pillar to prove his son wrong, it split open and Narasimha emerged: half-man, half-lion, satisfying every condition of the boon at once. Narasimha killed Hiranyakashipu at twilight (neither day nor night), on the threshold of a doorway (neither indoors nor outdoors), across his own lap (neither ground nor sky), using his claws (not a weapon).
Why the story still matters
The story is usually read as a lesson about the limits of power gained through boons or cleverness when it isn't matched by righteousness, and about how sincere devotion (bhakti) can outlast even a parent's authority when the two are in direct conflict. It's the founding story behind Holika Dahan, the bonfire lit the night before Holi, and Narasimha Jayanti is celebrated separately in his honour.