Sayian when reach to certain powerlevel or when is at the extreme point of its anger,it transform into the next stage.
Its just like evolution of Pokemon.After transforming,it gets new powers with ultimate speed.
There are five Stage of sayain transformation:
Base form :Every sayian at the first stage of transformation
Super Sayian :Goku was the first to become it.Every mian protagonist became Super sayian.
Ultra Super Sayian :Vegeta was the first to become this.Trunks and Goku later become Ultra Supersayian.
Super Sayian 2 :Gohan was the first to become SSJ2.Goku and Vegeta also reached to this stage.
Super Sayian 3 :Goku was the first to reach it.Gotenks also reached to this stage.
Super Sayian 4 :Goku and Vegeta reached to this stage first.
Super Sayain God :Only Goku is applicable to transform into this stage.
There are other classifications like False Super sayian and Legendary Supersayain but they are limited.
Goku is the only False Super sayian and Broly is the only Legendary Supersayian.