Pokémon of this type are specialists in melee attacks, such as punches and kicks. It could be said that they're the opposite to Psychic-type Pokémon because while Pokémon of that type depend primarily on special attacks, Pokémon of the Fighting type depend mostly on physical attacks. In particular, every Fighting-type attack introduced before Generation IV remained a physical attack, with special Fighting attacks only being introduced from Generation IV onwards.
Most Fighting-type Pokémon have a human-like body shape because they represent practitioners of various martial arts, which tend to be real-world humans. Some Fighting-type Pokémon are represented by looking like fighters (Machamp looks like a bodybuilder and Crabrawler looks like a french-wrestler) while other are represented by being based on a certain type of fighting style (Hitmontop is based on Capoeira fighting and Gallade is based on sword-fighting). They can also be represented by simply just having incredible strength (such Pokémon such as Bewear, Pangoro, Crabominable, Conkeldurr and others have this).
Fighting-type Pokémon have varied ways to evolve as this represents a fighter going on a journey to get stronger and find out who he/she is supposed to be.
They're weak against the Psychic type because sayings like "Brains beat brawn" and "Mind over Matter." The brain also regulates all muscle functions in most animals.
They're weak against the Flying type because it's been long said that height is a large advantage when it comes to battle, as it's difficult to hit something that's moving in the air and that dropping attacks have the advantage of gravitational acceleration when it comes to attacking.
They're weak against the Fairy type because physical strength can't do anything against magic.
They're strong against the Dark type because it's a common theme in martial arts and fighting fiction that goodness always wins, and a fighter must be prepared to face his/her fears, some of which are represented by the Dark type.
The Poison type resists them because the body is a fighter's weapon, and it can be damaged by poison.
The Bug type resists them because some bugs and insects can resist punches easily. Also, it is difficult to hit an insect because it's small and moving quickly.
They're strong against the Ice, Rock and Steel types because ice can be shattered with a strong punch, some fighters are trained to destroy rocks, and steel objects can be easily bent.
They're strong against the Normal type because a fighter can easily beat a normal person.
The Ghost type is immune to them because it's impossible to hit an incorporeal being. Interestingly, they're not immune to the Ghost type.