The first class, also known as 'the elite' find themselves better than the rest of the Saiya-Jin, they are very strong and expect that they will be the first ones to reach the legendary Super Saiya-Jin level. The most important Saiya-Jin in this class are King Vegeta, Queen Cerori and their son Prince Vegeta.
The first class also consists out of the elite fighters, which usually act as the guards of the Royal children or second class children, even when they can defend their own. The guards are very dedicated to these children and they will sacrfice themselves for them.
The second class is a bigger class than the first, but a smaller one than the third. These are the average Saiya-Jin. Their children are thought to still have potential, especially for breeding and so they are kept safely.
The third class Saiya-Jin are just the 'common soldiers', used to take over the regular planets without any particular difficulties. They are all very rowdy and less refined than the other two classes and also have more children running around, which is rather nasty since the first and second class children are the ones that are used to breed. This is the biggest class with the majority of the Saiya-Jin.