A Martial Artist is a warrior who fights by only using their body and attacking just with fists, kicks, and throws. They are all users of unarmed martial arts, unlike swordsmen who choose blade weapons for combat and snipers who choose guns.
The martial artists shown in the world of One Piece, like swordsmen, are an important asset to a crew. Martial artists are often found in positions of high respect, just on par with their respective swordsmen. It should be noted that in the One Piece world, there are two main types of martial arts; free-form and style-form.
Free-style fighters, who use just their bodies, are considered martial artists but would be considered 'brawlers' or 'street fighters' in the real world, and not a martial artist. An advantage of free-style is adaptation, free-stylers are not bound by the forms and exercises of style-base fighters.
In comparison, the 'style-based' martial artists in the real world would be practicing the true martial arts. They use traditional stances and attacks that can be identified just like attacks in the real world can identified as Judo or Karate. Fishman Karate has different stances than the kick-based style that Zeff and Sanji uses.
The main martial artist of the Straw Hat Pirates is Sanji (although he's not the only one who fights unarmed). There exist many martial artists with different fighting styles. In the One Piece world, most martial artists have in their arsenal an inhuman strength and speed (and in some occasion Devil Fruit abilities that enhance their power). Their fighting styles can be martial arts with conduct codes or simply improvised styles with no codes of combat.