In Japanese, "jin" means people; so that means that saiya-jin are "Saiya people". Their sole purpose in life was to fight and survive. At the beginning of the Dragonballz series, we learn that they lived on Planet Plant. They shared this planet with a race called Tsufuru. The Tsufuru were an advanced race with many cities and highly-advanced technology. Their technology allowed them to fight against the naturally strong saiyans. One night on the full moon, all the saiyans who had been exposed to the light of the full moon, underwent the change into Oozaru (Giant Ape) and easily crushed the tsufurus out of existence. From then on the planet was called Planet Vegeta. The saiyans used some of the tsufuru technology to become the strongest race in the universe. The strongest being in the universe, Frieza, an evil creature, used the saiyans as mercenaries to take over planets for him and sell them. One day, the saiyans got tired of working for Frieza, so the king, King Vegeta, revolted with several of his loyal men. Unfortunately, he was killed along with his men. Frieza then destroyed Planet Vegeta in one giant blast, along with almost the entire saiyan race. The only survivers were Kakarotto (Goku), Prince Vegeta, Radditz, Nappa, Taurus, Paragus and Brolli.
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(Normal) Saiyan:

This is the typical appearance of a saiyan. A normal saiyan has black hair, a thick, strong, muscular body, and a unique tail. Saiyas have the excellent natural fighting ability that causes them to consume a great amount of calories daily. In order to provide these calories, they eat a lot. Compared to humans, saiyans are extremely strong. Even when hit by Bulma's car in the beginning of the Dragonball series, Goku was not hurt. He immediately stood up after he was hit.

Saiyans have the super ability of changing (upgrading) to stronger forms, as described below.

Super Saiyan 1

Goku is the first one we ever see doing it. In this level the Saiyan's hair turns goldish blond, the eye's turn green, and his power level rises greatly. At first, the Super Saiyan 1 level is usually triggered by emotional rage but once reached they can go SSJ1 at will.

Those who achieved this form: Goku, Gohan, Vegeta, Trunks, Goten and Brolli

Ultra Super Saiyan

This stage is between SSJ1 and SSJ 2. During this stage the Saiyan's muscles greatly increase in size and they become very strong but very slow. When Goku and his son, Gohan were training in the Room of Gravity and Time during the Cell saga, Goku demonstrated this stage to Gohan, and told him that it was useless because of it's drawback of making a saiya-jin usually slow.

Those who achieved this form: Goku, Mirai Trunks, Brolli, Vegeta

Super Saiyan 2

This stage was first reached by Gohan while fighting Cell in the Cell Game. This stage is very much like SSJ1, but of course much stronger and faster. Occasionally, the Saiyan who is in the form has electricity going around his body and through and they have slightly longer hair. It is initially triggered by extreme rage.

Those who achieved this form: Gohan, Goku, Vegeta, Brolli

Super Saiyan 3

SSJ3s have golden hair which gets very long and full and their eyebrows disappear. This level is only achieved through training. After training in Heaven after the Cell saga, Goku learned how to transform to SSJ3. This form brings more power and speed but uses more energy so this form can only be sustained for a short time.

Those who achieved this form: Goku, Gotenks (Fusion of Trunks & Goten), and Vegeta

Super Saiyan 4

In this form the Saiyan's tail grows back and their eyes turn yellow with a red outline around them, their palm contains a hole that can absorb energy, their back and most of their body is covered with red fur, their hair is black but very long, and they get unbelievable strength and speed. Goku is the only real person to reach this level, but Bulma creates a device that turns Vegeta into a SSJ 4 also. For this fact it is believed that this is the true from of a super saiyan because only one person could reach it and that was a true blooded saiyan. This form is triggered by extreme rage, anger and sorrow.

Those who achieved this form: Goku and Vegeta

Fusion

The following are the Three ways to Fuse.

Fusion Dance: Basically just what the name suggests, a dance in which two fighters perform a dance to fuse into one whole being. Goku learned this style of fusion during the five years of inactivity between the end of the Cell Game and before Gohan's start at Orange Star High School. The fusers must be similar in size, build, and power level (The stronger fighter can simply just lower his power level to match the fighter with the lower power level to meet this requirement). Any mistakes in performing the dance can cause defective fusions. This form of fusion only lasts for exactly 30 minutes, so after this time period the two fighters de-fuse into their own two beings. It hasn't been successful to break the fusion before or after the 30 minute time period.

Requirements/Conditions: - The two fighters must be similar in size, build, and power - Must perform the dance correctly for a successful fusion - Fusion only lasts for 30 minutes

Porterra Fusion: The fusion technique of choice for Kaioshins, this technique is allegedly hundreds of times better than that of the "Fusion Dance". The two fighters that wish to fuse had to wear an earring, each having one in the opposite ear of the other fighter. After this is done, they can fuse into one being (Which in prospective causes the final being to have 2 earrings on). This fusion was meant to be permanent, but unexpected conditions caused the break-up of the two fusers.

Requirements/Conditions: - A set of earrings (one per fighter) - Both fighters must wear their earrings on opposing ears - Each person can ONLY have one such fusion in a lifetime - The effects of the fusion is permanent (supposedly)

Namekian Fusion: The way of fusing by the Namekians. This is a way for 2 Namekians to fuse into one being. No special objects or performances are needed for this type of fusion. The fuser just has to place his hand on the person that he wishes to fuse with (being that the person being fused allows the act), and they just merge into one. There are no set time periods for this type of fusion, and the two fighters can de-fuse at any point of time.

Requirements/Conditions: - The fuser's must be Namekians (makes some sense?) - Both fusers must agree to fusing to each other - Size and Power level has no effect on the fusing

Examples of Fusion are Gotenks, Gogeta, and Veggeto. Piccolo and Nail, plus Piccolo and Kami

Oozaru

Oozaru is the Japanese word for "Big Monkey". When a Saiyan with a tail is exposed to pure moonlight (from any moon) he turns into a gigantic 20-foot tall, ape-like creature. The Saiyan's power level is increased ten times in this form. Saiyan's used this form to take over planets for Frieza. This form is possibly the most powerful form under SSJ level 3. All saiyans, except Goku and Gohan were able to control themselves during this form.

Those who achieved this form: Goku, Gohan, Vegeta, and other saiyans that took over planets for Frieza

Ougon Oozaru (Golden Oozaru)

This stage is much like the regular Oozaru stage except that the fur is golden not brown and it's of course much stronger. To reach this stage a Saiyan must have a tail and must receive more waves from the moonlight. The only ones to ever reach this stage were Goku and Bebi Vegeta during Dragonball GT. Vegeta reached it while Bebi was inside of him and attacked his friends. He was eventually defeated by Goku.

Those who achieved this form: Goku and Bebi Vegeta.