Oozaru: I hpe I don't need to explain this one
however, the saiyan becomes a giant ape, this form is only obtainable if the saiyan still has a tail.
obtained by: Goku, vegeta, radditz , nappa, gohan , pretty much all other saiyans , but not by the generation after gohan.
Saiyan: Saiyans have tails and when reached a certain powerlvl they can transform to ssj & beyond, also when they see a full moon, they can transform to oozaru. One other note All have black hair, there is a glitch in the series, since in the first few dbz episodes vegeta has red brownish hair, however this is fixed later on.
obtained by: All
SSJ: The saiyan gets more bulked up, he gets faster stronger, the color of his aura changes, and (not unimportant) his hair turns yellow/goldish this level is only obtainable when reached a certain powerlevel and when certain emotions occur (now don't ask me how goten and trunks did that).
Obtained by: Brolli Goku Vegeta Gohan Trunks Goten Pan (goku and vegeta jr?)(probally more in gt)
ASSJ: This is the form trunks and vegeta achieved when they came out the hyperbolic time chamber. The saiyan gets more bulked up, and his speed drasticly decreases, on a side note, his power drasticly increases . However the total of both is closer to SSJ then SSJII. The hair gets a bit more spikey then before.
Obtained by: Goku, Vegeta, Trunks, Brolli (not sure if he skipped it)
USSJ: This is almost the same as ASSJ, however, there are differences for one thing, USSJ is bulked up even more! The speed is decreased even more oo and the power increased more too. One major difference is that the pupils can not be seen, thus the saiyan has no puils, his hairs are even more "spikey". This form however is more close to ssjII because the total of speed and strength is closer towards it.
Obtained by: Trunks, Brolli
LSSJ: Legendary super saiyan , this form is only abtained by brolli, and looks a lot like ussj, the only difference there seems to be is that he has no speed downfall , his speed did not decrease wich gave him more of an advantage over ssj. This form is still weaker then ssj2 tough.
Obtained by: Brolli
Perfect SSJ: Not a real form, however, not something to be missed. This is the stage where the SSJ has mastered its form. His power doesn't increase here, however the form does not drain power any more, he can transform instantly when he reached this stage
Obtained by: Gohan Goku Vegeta Trunks Goten
SSJII: This is the form after SSJ, it's an upgrade over ssjII, he becomes bulkier (as in his muscles grow) he becomes faster, his power lvl drasticly increases too!! His hair is flowing upwards except one hair at the front (not see-able with vegeta) and elictricity is going over the body. This form is only obtainable when reached a certain powerlvl and emotions occur
Obtained by Goku, Vegeta , Gohan eventually the younger saiyans too.
SSJIII: The form that surpasses ssj2, the saiyan has no eyebros any more, and his hair grow insanly long, he gets stronger and faster, his powerlvl increases a lot. This can only be obtained after heavy training
Obtained by: Goku, Gotenks (fusion)¨.