MMANiagara.com

basic MMA only breaks the game down into 3: Stand up(boxing, kickboxing/Muay Thai and forms of full contact karate), Clinch(Freestyle, Greco-Roman wrestling, Sambo and Judo) and the Ground(Brazilian Jiu-Jitsu, shoot wrestling, catch wrestling, Judo and Sambo).

but since you are asking for styles, UFC enumerates the following arts (though I personally think there are more styles/techniques that are utlilized in MMA today):

Boxing
The skill or sport of fighting with the fists usually with padded leather gloves. Referred to as the "sweet science," boxers use elaborate foot maneuvers and quick jabs for offense.

Brazilian Jiu-Jitsu
In the mid-1920′s, Carlos Gracie opened the Gracie Jiu-Jitsu Academy in Rio de Janeiro, Brazil. He taught the skills he learned from Japanese Judo master Esai Maeda. The skills were later modified to use less strength and to be more effective against larger opponents. Brazilian Jiu-Jitsu’s reputation spread due to the success of its practitioners in no holds barred contests.

Freestyle Wrestling
Possibly the world’s oldest sport. Contestants struggle hand to hand attempting to throw or take down their opponent without striking blows. Some of the many styles of wrestling are Freestyle, Greco-Roman, and catch as catch can.

Greco-Roman Wrestling
Possibly the worlds oldest sport. Contestants struggle hand to hand attempting to throw or take down their opponent without striking blows. Some of the many styles of wrestling are Freestyle, Greco-Roman, and catch as catch can.

Jiu Jutsu
Ancient Japanese martial art that encompasses throwing, joint locks, striking, and weapons training.

Judo
Sportive Japanese martial art founded in 1882 by Jigoro Kano. Derived from Jujutsu, Judo is now an Olympic sport that emphasizes throws. Striking is not allowed in competition Judo.

Karate
Name used to identify many Japanese and Okinawan martial arts. While known for powerful, linear techniques, many Karate styles also incorporate softer, circular techniques. Some of the popular styles of Karate are Kyokushinkai, Shotokan, Goju-Ryu, Shorin-Ryu, and Kenpo which was the first "Americanized" version of Karate.

Kickboxing
Sportive martial art combining boxing punches and martial arts kicks. Many different styles with different rules exist such as Muay Thai, Full Contact Karate, and Asian Rules Fighting.

Kung Fu
Also referred to as Gung Fu, Chinese Boxing, and Wu Shu. There are hundreds of Kung Fu styles. Many are patterned after the movements of animals. Some well known styles of Kung Fu are Wing Chun, Praying Mantis, Pau Kua, Tai-Chi-Ch’uan, and Shuai Chiao.

Tae Kwon Do
One of the most practiced martial arts in the world, Tae Kwon Do is a Korean style known for its flashy kicking techniques.

Wrestling
Possibly the world’s oldest sport. Contestants struggle hand to hand attempting to throw or take down their opponent without striking blows. Some of the many styles of wrestling are Freestyle, Greco-Roman, and catch as catch can.