Saiya-Jin

Saiya-Jins are all born with tails and are able to transform with the help of the full moon.  A tail will grow back only when the Saiya-Jin is a child.  A pure breed hair dose not grow from birth and once they reach adult their physical appearance does not change much.  Anything less then a pure breed Saiya-Jin their hair can grow and will continue to age as they get older but still slower then a normal human.  Pure breeds are Vegeta, Goku, Raditz, Nappa. There are some more pure breeds but they are only found in the DBZ movies not in the Series.  Half breeds are Gohan, Goten, Trunks, Bra.  The only quarter breed is Pan.  All Saiya-Jins get stronger with every battle they fight which makes them the supreme warrior race of the universe.

Oozaru

Any Saiya-Jin with at tail can transform to this state as long as they have the full moon to help them.  This transformation increases their power by 10 times.  This transformation is only to be use by skilled adults.  If children transform them are unable to control their power and just go on a rampage.  Used effectively by Vegeta, but also Goku and Gohan both transformed when they were children.  

Super Saiya-Jin

Super Saiya-Jin was a legend Vegeta remembered that took place over 1000 years ago.  This Saiya-Jin had powers beyond anyone's wildest dreams but so much power drove the previous SSJ insane.  To first become a SSJ you need to have 3 things.  A pure heart, extreme rage, and to be very strong.   It seems though that the half breeds have and easier time becoming SSJ and can do it at an earlier age too.  All Saiya-Jins can become SSJ but the reason that the Females never change is cause the creators could not think of how to make a girl look SSJ and that's the truth.    SSJ's are Vegeta, Goku, Gohan, Trunks, Goten.

Super Saiya-Jin Full Power, Ultimate SSJ

This is that state that is almost breaking the barrier of SSJ2.  Same as normal SSJ but this USSJ has huge muscles making him a lot stronger and a little faster then normal SSJ.  Future Trunks and Brolly are the only ones that actually used this form to fight but Goku, Gohan and eventually Vegeta could use it too. 

Super Saiya-Jin 2

The level that exceeds a SSJ is them same except for larger muscles, their hair stands straight up after growing a slight bit, and a electric aura surrounds them.  This greatly improves the Saiya-Jins speed and power far beyond even the USSJ.  Gohan was the first to use this form when fighting Cell. Vegeta and Goku both used it when fighting each other in the Buu Saga.  To become SSJ2 is not and easy thing to do because you need to have more extreme rage just like you need to change to SSJ. 

Super Saiya-Jin 3

The transformation the is beyond the level that exceeds a SSJ.  This is considered to be the last true transformation by many Dragon Ball Fans because it was that last form that actually looks like what we call a Super Saiya-Jin.  When Transformed to SSJ3 you hair grows very long, more then 3 times in length.  The eyebrows of the Saiya-Jin totally disappear and the face seems more like a villain as Gotenks states after he transforms.  Goku is the first to show off this awesome power while fighting Buu.  Gotenks quickly learns SSJ3 after Trunks and Goten learn the Fusion-Dance.  Vegeta Eventually goes SSJ3 in the GT series with the help of Bebei.  

Golden Oozaru

This form is only motioned once when Vegeta is talking about the Legendary Super Saiya-Jin but no one is able to reach this state in DBZ.  In DBGT Goku was the first to change to this while fighting with Bebei.  This is just like that Normal Oozaru but gone SSJ.  The golden aura of the SSJ makes that Oozaru look like a golden ape.  This is like a double Transformation 10 time more power for the Oozaru and then more power from the SSJ.  Goku was never able to control himself in the transformed state but learned quickly when he had too.  He did not do much fighting in this state but it enable him go get to the next form instead.

Super Saiya-Jin 4

Goku and Vegeta are the only 2 that ever reach this state.  Maybe it is because they are pure breeds or cause everyone else quit training and they never stopped.  In the form that hair is no longer golden but back to its original color.  Their chest and arms are covered in thick fur so it looks like they are wearing a shirt but they are not.  Power at this stage is immeasurable but they still always manage to find and enemy that is just as strong.  There is one more SSJ4 I forgot to mention, SSJ4 Gogeta.  Fused using the special dance Gogeta fight against the 7 Evil Shenrons at the every end of GT.  Even thought still SSJ4, Gogeta manages to have RED hair.  I don't know how that work but he does.  This is that last of all transformations in any of the DB, Z, GT , Movies, and Specials.