(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. Saiyans 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. All full-blooded Saiyans share several common features: All are born with the hairstyle they keep through out life all are very strong and muscular and they all have a tail. The tail is a weakness for weaker Saiyans, as any pressure applied to it is temporally paralyzing, but Saiyans who have strengthened their tail, are not affected at all by this. The tail is also necessary for the transformation to the Oozaru state. Also all Saiyans have a strong love to fight, which can sometimes be a weakness.
 

Giji (Half Super Saiyan)

This is a mid point between a Saiyan's regular from, and Super Saiyan. Goku reached this form in DBZ Movie #4, when he was really mad at Lord Slug. This is where the Saiyan can almost become a Super Saiya-Jin, but is lacking one or two things.
Achieved By: Goku

Mind controlled Super Saiyan

This is the weakest of all Broly's forms and is a slave to his father Paragus, thus he cannot fully maximize his Super Saiyan potential. This form is first awakened by Goku when Broly first sees him in DBZ Movie #8. However, this form does not last when Broly breaks free of Paragus' control. When Broly is free he is able to release his full power.
Achieved By: Broly

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 - 1 (dai ni dankai)

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, Vegeta

Ultra Super Saiyan - 2 (dai san dankai)

This stage is achieved by goku and trunks. in this form the muscles are bulked up heavily giving the person incredible strength far more than ultra super saiyan 1! but at the cost of speed as trunks realised painfully against his battle with perfect cell. goku and the others refrained from using this form.

Those who achieved this form: Goku, Mirai Trunks

Legendary super saiyan

this is similar to ultra super saiyan 2 but far more unique as it is achieved only by one saiyan. in this form there is great power far more than ultra super saiyan 2 as the warrior can retain his speed! this form has only been achieved by broly.

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

Mystic Saiyan

This form is achieved by Gohan after his power up from Old Kai. During this state his power is fully maximized. So much so that he does not need to ever go Super Saiyan to increase his power. It looks exactly the same as his normal state, as you can see from the picture. If the Old Kai had finished the power up Gohan would have been stronger than SSJ3 Goku, but he never finishes because he is too lazy.
Achieved By: Gohan

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)

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

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.