In the very early stage of a pure-blooded Saiyan-Jin, they all have black hair and they look just like ordinary people, except they have a tail at their rear end (like a money). They gain their power level by training, fighting and improving. The maximum power level can be gained by an exceptionly powerful Saiyan-Jin is around 20,000 or 30,000. However, another way is to gain power by looking at a full moon (Ozuro Mode). By looking at a full moon will transform them into a giant monsterous ape, which their power level is increased 10 times.
Advantage: When a Sayian-Jin is turned into an ape (Ozuro Mode), they gain great stength.
Disadvantage: If the moon is suddenly covered up by dark cloud or obstructed by a certain object, the Saiyan-Jin ape is instantly instantly transformed back to a normal Saiyan-Jin
The first Super Saiyan-Jin appeared in the DragonBall serie, there Gokou was fighting against Freeza. To transform into a Super Saiyan-Jin, the Saiyan must be able to build up the anger inside. Let the anger to increase the desire of revenge therefore pushing the Saiyan-Jin's power level suprassing the maximum limit. If the process is successful, the Saiyan-Jin should have gold hair which sticks up and eyes turning green. They also have this glow of golden light wraps around the Saiyan-Jin. As a Super Saiyan-Jin, it increases the power, speed and agility. This was how Gokou killed Freeza in the DragonBall series.
Advantage: It increase the overall power.
Super Saiya-Jin 2:
Again, to get to this stage, a Super Saiyan-Jin must be able to push past his/her own maximum limit. To get to this process is much easier than transformation from Saiyan-Jin to Super Saiyan-Jin. Just by working extra hard, any Super Saiyan-Jin can get to this level. A successful transformation will result in gold solidified hair which is split into pieces of triangular strips. The hair still sticks up and the glow of golden light is still around the Saiyan-Jin.
As a SSS 2, they have the abilty to adjust their strength/speed/agility. More experienced fighters such as Vegeta and Gokou, they know how to balance up to make them well adjusted. If successful, it can also result in increasing more strength/speed/agility.
If one part of the power goes up the other two have to go down. For example, if a Saiyan-Jin chooses to have more strength, he/she has to sacrifice speed and agailty. Simple as that.
Super Saiyan-Jin 3
This is once again, the next stage from a SSJ 2, or possibly the final transformation. By expanding the power (similar to the transformation from Saiyan-Jin to Super Saiyan-Jin) to get to this stage. However, getting into stage isn't easy, it requires MUCH MORE strength to transform from SSJ2 to SSJ 3. The only difference between a SSJ 2 and SSJ 3 is that SSJ 3 has much longer hair and a much more vicious look. The eyebrows are gone and overall they look very cool. They don't look very muscular, a pretty much the same as SSJ 2, but the hidden power is something that cannot be compared.
The SSJ 3 has much greater strength/speed/agility. Which it makes the Saiyan-Jin as a true complete fighter.
Since a SSJ 3 is much more powerful, it uses up more strength too. A SSJ 3 fighter can fight for a while, but he/she gets tired very quickly too. From Gokou's final against Majin Buu, we know that resting will not build up or regain strength, it losees them.
Super Saiyan-Jin 4
SSJ 4 is the last and final stage of a Super Saiyan-Jin. This stage is called "Beyond Super Saiyan-Jin". To become a SSJ 4, the Saiyan-Jin must have a tail and plenty of strength (so the Saiyan-Jin can expand it's current strength and power to get to SSJ 4). Like the very beginning, when the Saiyan-Jin looks at a full moon, he/she is instantly turned into a Gold Oozaru, or a big monkey. While the Saiyan-Jin is in Oozaru mode, he/she goes berserk to expand it's anger and strength, hence becoming SSJ 4. SSJ 4 is not a big monkey, SSJ 4 is actually transformed back to a human like Saiyan-Jin. With long black hair, green eyes and red fur covering most part of the body except for the chest, and a long red tail. They still have that yellow glow around the Saiyan-Jin representing the Saiyan-Jin is now in SSJ mode, however, it is not a very distinct feature like SSJ, SSJ 2 or SSJ 3 since the glow is very weak.
It increases the overall strength/speed/agility. Which ultimately makes SSJ 4 greater fighters than SSJ 3. The strength, speed and agility are well balanced out, no need to readjust them to suit the Saiyan-Jin like SSJ 2.
It uses up too much power, just like SSJ 3. This transformation does not last long. After 10 minutes or so, the Saiyan-Jin transforms back to a normal Saiyan-Jin.