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. 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.
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, unusual and twisted "dark" minds can be unpredictable and incomprehensible and evil can corrupt the mind, and where ghosts are of a different frequency of existence and can annoy or play pranks.
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.