Saiyan - Saiya-jin
This is your average Saiyan. The most powerful technique a saiyan has is called Zankai, it basically increases their power greatly every time the are injured or is close to death, this has allowed low class saiya-jins to become 10 times stronger in just a few days making the Saiya-jins a great threat to any enemy.

Oozaru
The main features of this form is the fact that they become a 40ft tall giant ape, this form increases the Saiyans power by exactly 10 times. For a saiyan to go Oozaru they have to have their tail and they have to be exposed to a full moon, no matter the size of the moon.

False Super Saiyan - Giji Super Saiya-jin
False SSJ is exactly what it sounds like, it is "the false super Saiyan". This form was only ever attained by Goku in movie 4, he had the golden aura as usual, but no golden hair, think of this as half-way to being a super saiyan.

Super Saiyan [SSJ] - Super Saiya-jin
For a Saiyan to become a SSJ they needs to become angry to start the reaction. For example, when Goku went SSJ he was angry at the death of his friend Krillin. When future Trunks first went SSJ he was angry at the death of his life-long friend and mentor Gohan, at the hands of the androids. When they go SSJ he/she experiences a great increase in power, a large golden aura surrounds them, their muscle mass is increased and their hair goes a bright gold and stands on end.

Ascended Super Saiyan [USSJ1] - Super Saiya-jin Dai 2 Dankai
This is one stage beyond SSJ, the user gains a big increase in power but they suffer a side effect, they lose some speed. The first to achieve this form was Vegeta after he emerged out of the Room of Spirit and time [ROSAT].

Super Saiyan 2 [SSJ2] - Super Saiya-jin 2
SSJ2 is obiously the stage after SSJ1. It was first achieved by Gohan against Cell, Cell cruelly killed android #16 during his speech to Gohan. This enraged Gohan so much sending him to a new level of SSJ. When they become SSJ2, they receive yet another huge power gain, their hair has more visible depth, electricity is visible around their body and they gain even more muscle mass.

Super Saiyan 3 [SSJ3] - Super Saiya-jin 3
In my opinion, this is the coolest of the SSJ levels. When you go SSJ3 you grow long gold hair [as usual] but it grows down their back, they also lose their eye brows. As usual they get another great boost in their power. The first to go SSJ3 was Goku during the Buu saga, the biggest downside to SSJ3 is the enormous drain in energy they receive. After going SSJ3 for five minutes Goku was completely knackered, he had trouble staying on his feet.

Golden Ozzaru
A Golden ozzaru is basically a normal ozzaru just with Golden fur instead of brown. To go golden Ozzaru the user has to haev his Saiyan tail and instead of looking at a full moon he has to look at the Earth from a distance. A very weird way to transform, but that's just GT, all of GT was messed up. A golden ozzaru could match the power of a strong SSJ3.

Super Saiyan 4 [SSJ4] - Super Saiya-jin 4
This transformation is not even considered real by some. This is because all of the Dragonball/Z series was drawn by a man called Akira Tokiyama, he was also the creator and writer of Dragonball/Z. But he did not do anything for DBGT which was the only series where SSJ4 was featured. SSJ4 is definitely the weirdest looking of all the SSJ levels. When they go SSJ4 their hair remains black, no golden aura is created, and they grow red fur all over their body except their chest. To go SSJ4, first they obviously need sufficient power, they also need a saiyan tail or the reaction will not happen