Most notably, the common ones are:

Traditional Fighting Arts

These are systems such as Karate Do (Japanese & Okanowin), Kenpo (Chinese & Okanowin), Aikido, Kung Fu, Wing Chun, Tai Chi, and Taekwondo. They are well rounded and focus a lot on the "art" and "tradition" of the system. These systems have been around hundreds, if not, thousands of years and are taught pretty much the same has they have been.

Non-Traditional Fighting Arts

These are systems such as American Kenpo Karate, Jeet Kune Do, and Hapkido. They are also well rounded systems that focus on "tradition" and "art" like their older counter parts from which they where adapted from, but have been modified for our modern society.

Combat Self-Defense Fighting Arts

These are systems such as Krav Maga (Israeli), the Marine Corps Martial Arts, Target-Focus Training, and Special Forces Hand to Hand Fighting. These systems teach straight self-defense fighting with out any "tradition" or "art" aspects. They are hard core, no-non-sense, no rules self-defense fighting systems that are designed for the battle field. These are the kill or be killed fighting arts, no sport here and not for everyone.

If you have done any research into the different martial art systems to decide which system is right for you, you have no doubt come across articles that state "martial arts and self-defense is not the same thing", and in the most part this is true. The majority of martial art systems will teach you techniques and sparing, give you new belts because to can mimic back the techniques and katas, but not what it takes to survive real life attacks, hence "combat self-defense". This is where these arts separate themselves from the other martial art systems.

Grappling Arts or Sport Fighting Arts

These are systems such as Jujitsu, Brazilian Jujitsu, Judo, Muay Thai, exc… These are martial arts at the core but are more focused on the sport and competition aspect of the art and have rules for which the fighters must abide by.

Full Contact Fighting

These systems are fairly new and focus on actual full-contact fighting! The most common ones are Kick Boxing, and Mixed Martial Arts (MMA), most people know it as the UFC (Ultimate Fighting Championship) fighting. This is a mixture of the other systems, taking the best fighting aspects of them and combining them into one fighting system that best suites the fighter. These are more sport than Martial Arts and don't teach "tradition" or "art". Think of them more like boxing with martial arts techniques mixed in.

What Is The Best Martial Art For You?
That is a great question! But what you need to ask yourself is "what are you looking for?" "What do you want to learn?" and "Which Martial Art can give it to you"