In Hop Hop Hop Town, the local Pokémon had been turned listless by the psychic waves produced by Hypno. When a Drowzee nullified the waves, the Pokémon were cured, with the exception of a wild Psyduck that Nurse Joy had been looking after due to a chronic headache. Brock wanted to impress the woman, so he promised to take care of the duck-like Pokémon.
Later, it was discovered that Psyduck had a permanent headache. Not wanting the continuously ailing Pokémon, Brock ended up passing him off to Misty because she is a Water-type specialist. Misty tried to avoid the little thing, but she ended up capturing him by accident.
Out of all of Misty's early Pokémon, Psyduck is arguably the one with the most personality. He was heavily featured, and appears in many episodes, although his appearances started to grow sparce during the Johto saga.
Psyduck seems to truly love and care for Misty. Despite his faults, he always tries to be there for her, whether she likes it or not. Psyduck does not like to stay in his Poké Ball, and will usually pop out according to his mood. This personality trait has come in handy quite a bit- when Misty or her other Pokémon are captured, it is usually Psyduck who escapes and runs to get help.
Psyduck has trouble swimming, which is unusual for his species. He also has trouble in battle. He knows several powerful Psychic-type techniques, but can only use them when his headache is severe. Misty once considered trying to boost his power by slamming a bike onto his head.
Misty constantly complains about Psyduck's faults, but many suspect that she actually deeply cares for Psyduck- she has never considered getting rid of him, though it has been suggested several times; she instantly shoots down such ideas. Her antagonistic relationship with the little yellow creature is often compared to her relationship with Ash.
For more information on this Pokémon's species, see Psyduck.