The Psychic type is fairly common, and is very common among Legendary Pokémon. Psychic-type Pokémon tend to be very intelligent. It's also interesting to note that, many Psychic-type Pokémon are based on and/or related to real scientific or mythological discoveries such as Deoxys and Alakazam which are based on DNA and psychokinesis respectively. Many Psychic-type Pokémon also came from, are associated with, or have some connection to outer space, such as Elgyem which is based on common media depictions of aliens and Gothitelle which is said to have the ability to predict the future by reading stars.
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.
Famous Psychic-type Pokémon Trainers include Sabrina, the sixth Gym Leader in Kanto; Will, the first member of the Johto Elite Four; Tate and Liza, the seventh Gym Leaders in Hoenn; Lucian, the fourth member of the Sinnoh Elite Four; Caitlin, a member of the Unova Elite Four; and Olympia, the seventh Gym Leader in Kalos, and Tulip, one of the Gym Leaders of Paldea.
They're weak against the Bug and Ghost types and have no effect against the Dark type 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, and swarms of insects can kill concentration, where ghosts are of a different frequency of existence and can annoy or play pranks, and unusual and twisted "dark" minds can be unpredictable and incomprehensible and evil can easily corrupt the mind.
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. And skilled masterminds can overpower even the strongest fighters; within the realm of fighting, especially street fighting, it is often considered key to read the opponent and react appropriately in turn with how one defends themselves, regardless of strength.
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 (at least not in the literal sense). In addition to this, meditation expels impurities from the mind. This may also be because going into a medical field requires a lot of education and knowledge.
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 insusceptible to psychic powers. It may also be a play on the belief that tin foil hats can protect one's mind from psychic abilities such as mind reading or control.
They resist themselves because psychic people have trouble reading other psychics' minds.
Trivia[]
Along with the Water and Flying types, the Psychic type is the third type has been paired up with every type (excluding types such as ??? type and Shadow type).
Psychic is the only type to share a name with a move and a trainer class.