A martial art is any form of fighting and an art that has a set way of practice. A martial art is a style of combat, in many instances directed towards the self-defence. In the common usage, the word applies to the systems of combat developed in all the world.
The martial arts are fighting systems. There are many schools and styles of martial arts, but all share the same goal: self-defence.
Many martial arts include punches (boxing, karate), kicks (taekwondo, kickboxing, karate), holds and throws (judo, jujutsu, wrestling), weapons (iaijutsu, kendo, kenjutsu, naginatado, fencing, Filipino eskrima) or certain combination of these elements (several styles of jujutsu).
Martial arts are divided in two main sets: the so-called "hard martial arts" like karate and kickboxing which give special consideration to the attack to beat the opponent, and the "soft martial arts" like judo and aikido which fight the opponent in a less aggressive manner, using the force of the other to surrender him.
Martial arts are part of the art of war. If the main goal in a competition depends on noting points to somebody's advantage, then it could be said that this is a sport, not a martial art.