Through out the course of the series we see many different martial artists fighting with numerous different styles. Here I am going to attempt to explain a little bit about each of them.

Kame Sennin Style Martial Arts - This is the most fundamental style in the whole of Dragonball. It is taught by Kame Sennin to many of the major characters through out the series. One of its moves its the famous Kamehameha, which took Kame Sennin 50 Years to perfect, and Goku 5 minutes! His training consists of milk delivery, farm work, studying, swimming and construction work, all wearing 20 pound turtle shells!
Disciples - Son Goku, Kuririn, Gyu Mao, Son Gohan (Grandpa), Yamcha

Tsuru Sennin Style Martial Arts - This is the style of the Crane Master, Kame Sennin's rival. This is a very good style as its proved invaluable to Tenshinhan against Goku at the 22nd Tenkaichi Budoukai. Its main Ki move is the Dodon-pa, which resembles the Kamehameha, but shot from the finger. Nothing is known of his training methods
Disciples - Tenshinhan, Chouzu, Tao Pie Pie

Mutaito Style Martial Arts - This is the style of both Kame Sennin's and Tsuru Sennin's master, Mutaito. This is how the two masters gained their knowledge of martial arts, and also, I'm guessing, the knowledge of how to manipulate Ki, which they passed on. Again nothing is known of his training methods
Disciples - Kame Sennin, Tsuru Sennin

Karin Training - This is the training that anyone who can successfully climb Karin Tower receives. Part of the training is to climb the tower itself, but upon reaching the top, the objective is to take the Holy Water from Karin, which is meant to increase ones strength. This is just a pretext however, as the real training lies in becoming faster than Karin.
Disciples - Kame Sennin, Son Goku, Kuririn, Tenshinhan, Yamcha, Chouzu, Yajirobe

Kami-Sama Training - This is the style that is taught to Goku just before the 23rd Tenkaichi Budoukai. It increases his strength greatly. It is also taught to Kuririn, Tenshinhan, Yamcha, Chouzu and Yajirobe before the battle with the Saiya-Jins. The training consists of sparring with Mr. Popo, meditation and the Room of Spirit and Time.
Disciples - Son Goku, Kuririn, Tenshinhan, Yamcha, Chouzu, Yajirobe

Piccolo Training - This is the training that Gohan received prior to the Saiya-Jin fight. It constists of sparing with Piccolo. The method is very tough and demanding. As Piccolo will accept nothing short of perfection from anyone.
Disciples - Son Gohan

Saiya-Jin Training - This is the basic training that all Saiya-Jins go through. In the Bardock special we see Vegeta honing his skills against Saibamen. I assume that this is what Saibamen we created for, as training aids. Saiya-Jins also hone their skills by getting stronger after each battle.
Disciples - Vegeta, Nappa, Vegeta Ou, Radditz, Bardock, Any other Saiya-Jin not sent to other planets.

Kaio-Sama Training - This is the training that anyone who can successfully complete Snake Way receives. This is part of the training, as running 1 million kilometres is no small task. Upon arriving on Kaio-Sama's planet the gravity is 10 times that of Earth's so the training is very tough. The main moves are Kaioken and the Genki Dama. The main training is to catch Bubbles and hit Gregory with a hammer.
Disciples - Enma-Daio, Son Goku, Piccolo, Tenshinhan, Yamcha, Chouzu

Son Goku Training - This is the kind of training that Goku gave to Gohan in the Hyperbolic Time Chamber. The most fundamental aspect of this type of training focuses on how to become a Super Saiya-Jin. Goku pushes Gohan to the max during the training, making Gohan stronger than Goku himself at the end of the training.
Disciples - Son Gohan, Uubu

Son Gohan Training - This type of training gives the recipent a very good blend of styles that Gohan has learned. First of all there is Piccolo's style that Gohan first recived, then there is the training he recived form Goku and his own battle experiences on Namek and with Cell. This gives Gohan a very balenced style and this wealth of information is passed on to his students. Although it appears that Future Trunks gained more from Gohan than Goten and Videl did.
Disciples - Future Trunks, Son Goten, Videl