Aikido – Aikido is a Japanese martial arts style focused on redirecting the attack away from you. It involves grabs, strikes, throws, pins and locks. Steven Seagal, the movie actor, is a famous student of Aikido.
Bojutsu – A Japanese martial arts that is focused on staff fighting.
Boxing – Think Muhammad Ali and Mike Tyson. This fighting style is focused purely on powerful punches. There is no kicking or grappling allowed.
Brazilian Jiu-Jitsu – This Brazilian martial art focuses on ground fighting. This martial art exploded in popularity when Royce Gracie used it in order to dominate much larger opponents in mixed martial arts tournaments in the 1990s.
Capoeira – This Brazilian martial arts style focuses on kicks, knee strikes, punches and evading attacks. It is a very fluid and acrobatic martial arts (almost looks like dancing). Capoeira students are constantly moving in order to avoid attack or to set themselves up to launch an attack.
Eskrima – This martial art style from the Philippines focuses on the use of stick and blade weapons.
Hapkido – This Korean martial art focuses on punches, kicks, throws and joint locks. Hapkido classes often have some weapons training (i.e. with staffs and swords). Hapkido also emphasizes circular motion, non-resisting movements and control of your opponent.
Combat Hapkido is more focused on modern self-defense techniques than Hapkido. It eliminates some traditional Hapkido training elements (i.e. mediation and sword fighting) and adds more ground fighting techniques. It was created by an American, John Pellegrini in 1990.
Hanmudo – This Korean martial art is seen as a “smoother” and more “open hand” cousin to Hapkido. Hanmudo students also train with weapons.
Jeet Kune Do – This martial arts style was created by Bruce Lee (yes, the Bruce Lee of “Enter the Dragon” and “Fists of Fury”). This style focuses on techniques that are effective in real combat situations (versus martial arts that utilize stylized patterns or emphasize sports aspects or acrobatics). It focuses on kicks, punches, grappling and traps.
Judo – This Japanese martial art focuses on grappling, joint locks and throws. Punches and kicks are generally not practiced during Judo.
Jujutsu or Jiu-Jitsu – This Japanese martial arts style focuses on joint locks, holds and throws. It tries to redirect or manipulate the force of an attack in order to defeat the attacker.
Kajukenbo – This is an American martial arts that combines techniques from many different martial arts styles and was designed to be effective in real world self-defense situations and street fights.
Karate – This popular Japanese martial art focuses on punches, hand/elbow strikes, knee strikes and kicks. This style was highlighted in movies such as “The Karate Kid”. Wax on, wax off!!!!
Kendo – A Japanese martial art that focuses on sword fighting.
Kickboxing – As the name implies, this martial art combines kicks and punches (boxing). In American kickboxing, use of elbow and knee strikes is usually forbidden (unlike Muay Thai).
Krav Maga – This eclectic martial arts style from Israel focuses on winning in “real life” combat situations. It uses strikes, kicks, wrestling and many other techniques. Krav Maga wants to finish a fight as quickly as possible and therefore all attacks are aimed at the most vulnerable parts of the body (i.e. groin). This martial art is utilized by the Israeli Defense Forces.
Kung Fu (also known as Wushu) – This Chinese martial arts focuses on hand/arm strikes, kicks and even weapons training. There are many different schools and styles of Kung Fu (thus you can not “transfer” easily from one school to another if you move and this is one of the reasons that Kung Fu is not as popular as Taekwondo on a global basis). The best known Kung Fu students are Poo (of “Kung Fu Panda”) and David Carradine (of the “Kung Fu” TV series). 
Seriously, the Shaolin monks of China are famous practitioners of Kung Fu. Jackie Chang and Jet Li are probably the best known Kung Fu movie stars.
Shaolin Kung Fu is a well known style of Kung Fu. This martial arts was developed by the monks at the Shaolin Temple in China.
A well-known One of the more unusual forms of Kung Fu is Zui Quan or “Drunken Fist” style. This style tries to imitate a drunkard’s movements (i.e. staggering and falling over) in order to confuse opponents and allow for devastating counter attacks. Of course, the martial artist is not intoxicated when practicing this style of Kung Fu!
Kuk Sool Won – This is a Korean martial arts focused on strikes, kicks, grappling, joint locks, weapons training and healing techniques.
Military Martial Arts – These are martial arts used by and/or developed for the military. They are designed for real-life combat situations (versus sport-based martial arts that ban certain techniques in order to avoid injuries). Styles in this category include Krav Maga (used by the Israeli Defense Forces), Marine Corps Martial Arts Program (MCMAP), Combat Sambo (developed for the Russian military), Modern Army Combatives Program (US Army), Sanshou (developed for the Chinese military), etc.
Mixed Martial Arts – This martial art utilizes a variety of different martial arts styles (i.e. wresting, jiu-jitsu, boxing, karate, etc.) in order to defeat an opponent… most famously in the “Octagon”. The UFC is probably the best known promoter of mixed martial art events on a global basis.
Muay Thai – This martial arts style from Thailand is similar to kickboxing but also involves elbow and knee strikes.
Ninjutsu – This is a martial arts style developed from the techniques used by ninjas (Japanese spies and assassins). This style focuses on a variety of martial skills including unarmed combat, weapons, stealth, etc.
Sambo – This martial arts style was developed in Russia. Combat Sambo is similar to mixed martial arts and involves kicks, punches, wrestling, grappling and joint locks. Sports Sambo is more like wrestling and Judo.
Silat – This is a martial arts style that was developed in Southeast Asia (i.e. Indonesia, Malaysia, Thailand, etc.). Silat comes in many variations as each Southeast Asia nation seems to have put its own spin on this martial arts style. Silat focuses on techniques such as strikes (i.e. elbow and knee), throws, takedowns and weapons (i.e. kris sword and the karambit).
Taekwondo – This popular Korean martial art is focused primarily on punches, blocks, strikes and kicks. Many schools also teach joint locks and some throws. Taekwondo is best known for its spectacular kicks (i.e. tornado kicks and spinning hook kicks).
Tai Chi – This martial arts style is often seen as a “gentle” martial art because many seniors use its slow movements in order to improve their health & balance and to reduce stress. However, Tai Chi also has “harder” variations that include vigorous self-defense techniques.
Tang Soo Do – This Korean martial art is similar to Taekwondo and Karate (although it is not as popular as these two martial arts styles). Chuck Norris, the movie actor known for “Missing in Action”, “Delta Force” and “Walker, Texas Ranger”, is probably the most famous student of Tang Soo Do.
Wrestling – There are many types of wrestling such as Greco-Roman and Freestyle. Punches and kicks are generally not allowed. Wrestling focuses on grappling, throws and “pinning” your opponent.