Many Pokémon are of this type, and, being the type with most Legendary Pokémon, for many, the Psychic-type is the most powerful. Psychic-type Pokémon tend to be very intelligent. It's also interesting to note that, many Psychic-type Pokémon are based/related to real scientific or mythological discoveries such as DNA and Psychokinesis.
To attack, they use their mind power, thus many tend to have high special stats and low physical stats. In particular, every Psychic-type attack introduced before Generation IV remained a special attack, with physical Psychic attacks only being introduced from Generation IV onwards.
They're strong against the Fighting-type because there are sayings regarding the mind's superiority over the body ("Brain beats Brawn", "Mind over Matter," etc.). In addition, the brain also regulates all muscle functions. Without the brain, the body cannot properly function.
They're strong against the Poison-type because psychic concentration can at times ignore the body's physical pain. Also, while a person's body can be poisoned, their mind cannot.
They're weak against the Bug, Dark and Ghost-types because these represent fears, which can take over the mind and affect one's concentration, and of beings of different and hard to read psyches; insects have a much different mental process than vertebrates, unusual and twisted "dark" minds can be unpredictable and incomprehensible, and where ghosts are of a different frequency of existence.
The Steel-type resists them because some of them are based on machines, which don't have thoughts or feelings. As a whole they lack any mental capacity whatsoever, so thus are highly unsusceptible to psychic powers.