Martial Art Styles :

Aikido : "The way of strong spirit" (ie Combine Energy Way). They use Bokken quite a bit, and concentrate a lot on internal energy ('KI'). Japanese, although some could argue it is Chinese because a lot of internal styles originated from Daoism. Aikido style involves these - diverting strikes and move in harmony along an opponent, dynamics of movement, breakfalls, strikes to pressure points, Chi energies, calmness and control etc.. 

Arnis : A filipino art involving knife and short-sword techniques. 

Ba gua (or Pa Kua) : Chinese boxing involving a lot of internal work and spiritualism. Practicioners "walk the circle", and it has a lot of elements dealing with the Trigrams. Also known to some as "8 trigram boxing". 

Brazilian Jiujutsu : The style of fighting that the Gracie family developed. Basically, it is brazilian wrestling. Royce Gracie has won the UFC (Ultimate fighting championship). 

Escrima : Filipino art including use of stick strikes to joints and locks. Bruce Lee has adopted  this art in his fighting and proved it to be very effective system for any martial artist.

Fencing : Sword duels between practicioners using a thin weapon called an Epee (Pronounced Uh-pay). Points are scored when you touch the opponent with your Epee (Epee's are also called Foils), and practitioners must call out "Touche" if they are touched. 

English Boxing : Known as boxing to most. Involves learning footwork and punching. Some popular people in Boxing are Mike Tyson, George Foreman, Evander Holyfield etc. 

GUNG-FU: The Cantonese term for Martial Arts. See Wushu.

HwaRang-Do : Is the art of the Korean warrior divided into four powers and has a different kick for every day of the year. Mind control, Weapons, Healing and Strength are a major part of this system. The most famous of these warriors were call the Sul Sa who were considered a one man army by many.

HAPKIDO : A Korean art that is a combination of Aikido and Taekwon-do. A Hapkido artist will hold his ground until attacked then overwhelm him with kicks and punches. There are little forms or Katas in Hapkido and is strictly for self defense. Known as the way of coordinated power and consisting of three main principles: No resistance of force, Countering with circular motions and penetration of attackers defenses.

Iaido : Japanese sword art. Deals mostly with sitting meditation, unsheathing the katana, and cutting before returning to the sitting meditation. 

Judo : Japanese equivalent of English wresting. Also deals with joint locks, etc. An Olympic sport. Its throws and locks are devastating and can Kill !.

Jujutsu/Jiujitsu : Traditional Judo (Anything with the suffix Jutsu is an older art, with less competition. ie Its not competition oriented and use strictly only for real self defense ! ). 

Jeet Kune Do : Art developed by the late and great Bruce Lee. Emphasis is more on being versatile rather than using set patterns as in other arts. "Style of no style", "Art of the intercepting fist".

Kendo : Competition Kenjutsu. An excellent way to start learning the art of sword fighting !. As you can guess a sword is considered to be the most powerful weapon.  

Kenjutsu : Japanese fencing. Practitioners use Katanas (More commonly [and incorrectly] called Samurai Swords) to try and cut an opponent. Its simply is the art of swordsmanship.

Kali : Filipino art involving countering and  footwork. 

Karate/ Karate-do : Japanese art involving almost everything. Practitioners use set patterns of attacks .  Karate is considered to be the most refined martial art in the world that exists today. Karate is the most effective system, of self defense because it uses only empty hands. Just think your hands, legs and body working like a weapon that  can kill !. Even though Karate uses a fixed set of patterns/stances it can be deadly in real fights !. In a real combat/fight Karate student must  be flexible, skilled enough to change the rigid stances/patterns into natural and fluid moves/forms. Often you can see this transition in Karate's kumite or sparring.

Ninpo : Supposedly teaches how to be a "ninja". Only very few exists today. They are supposed to be in an higher order than a Ninja !

Ninjitsu : Traditional Ninja art. Ninja secrets were never given out....True ninjitsu (the way ninja 500 years ago did it) is dead. Once again, don't fall victim to claims of people teaching this. True ninjitsu isn't really even a martial art, it is more like "How to kill people undetected" (Hey, it was Feudal Japan). Some schools teach its more useful tactics and skills which can be very useful.. Skills of a Ninja is considered to be the most efficient and supreme !

Shaolin Arts (Wing Chun, Jun Fan, Five-Animal Kung Fu): Developed in the Shaolin temples of China, these arts were used to protect the monastery and keep the monks healthy . Movements are long, flowing and continuous, and provided Shaolin practitioners with the equivalent of t'ai chi or pakua. These systems were taught to some extent to all monks as part of their training. The complete systems were reserved for the few, the priests that would remain in the temple after being granted their priesthood. Most of today's martial arts can find their roots in Shaolin Martial arts. Even Karate has evolved from Shaolin temple. 

Tae Kwon Do : Korean art involving lots of kicking and other techniques. Board breaking is an important part of training.. More emphasis is on Kicking and the competition fights allow only kicking. 

A method of defending yourself with your hands and feet. 
Tae - jumping or flying, to kick or smash with the foot. 
Kwon - to punch or destroy with the fist.
Do - the "art" or "way" of.

Today this is more a competition oriented and an Olympic sport. 

Taijiquan (Tai Chi C'huan): Chinese internal art which involves many slow and flowing techniques which promote healthy Chi. Practitioners usually wear Kung Fu uniforms. They use less physical or athletic exercises and use the cultivation of  Chi. I know that the Taijiquan routines are worthwhile, but you would probably be better off learning from a traditional master... and give importance to physical fitness too. Its breathing and movements are considered to be the best and develops physical and mental balance.

these are some other styles not all though, there are many