Early Bird halves the sleep duration. When the Pokémon is put to sleep, the number of turns it will remain asleep is preset, between 1 and 5 (the range depends on the generation). This number of turns is halved for a Pokémon with Early Bird, rounded down if it is odd. So if only 1 turn is preset, it is rounded down to 0, causing the Pokémon to wake up the next time it moves.
This also applies to the self-induced sleep from Rest, reducing its length from 2 turns to 1 turn asleep (not counting the turn in which Rest is used).
Outside of battle
Early Bird has no effect outside of battle.