The regular Saiyan is just like anyone else. Well not really he (or if your Pan she) is stronger, and they would have a special ability to get stronger. How this works is if the Saiyan is at a near death situation then gets revived the power of the Saiyan will greatly increase. The Saiyan can also fly, form KI blast, and all the other stuff DBZ people do.

Oozaru (Full Moon) - Golden Oozaru

This stage of a Saiyan happens only to the Saiyans with tails. For this to occur you need a full moon, hense the Funimation name (Full Moon). After the Saiyans get their tails cut off it's unable to be performed. Basically all that happens is you turn into a big monkey. This increases strength, and doesn't lower speed too much. In Dragonball Z only Gohan and Vegeta actually perform it, but in Dragonball and GT we see it a bit more often. Golden is just the same as normal except your in the Super Saiyan form. Only Goku and Bebi Vegeta reached this level. This is a very strong Saiyan level.

Super Saiyan

The Super Saiyan level was first achieved by Goku after Freiza killed Krillen. If a Saiyan wants to reach the Super Saiyan level, he must have a need to transform to it. For example if everyone was being killed by Cell. The Saiyan must also have a clean/pure heart and a very high power level. When a Saiyan reaches this level his hair turns gold/blonde, he has a golden glow around him, and his strength and speed increases. Goku, Vegeta, Gohan, Goten, Trunks, and the fused characters all reach this level.

Super Saiyan II

The SS2 level can only be reached by anger. When a Saiyan reaches this level his hair grows a little more, sticks straight up, and one little hair strand thing hangs down. Also the glow around increases having electric waves also. The mian achievement of this level is that your speed greatly increases and you get stronger. Before Gohan discovered this level the Saiyans could only become extremely strong but lose their speed. Only few Saiyans reach that level.

Super Saiyan III

For a Saiyan to reach this level he would have to train very much. Very few Saiyans reach this, Goku is the only non-fused Saiyan to reach it. The big visual change in this stage is the hair, it grows down to the knees. Also the eyebrows disapear. Speed and strength once again grow very much. The body of a Saiyan can only take this for a short while the level decreases the power so much. Gogeta and Gotenks reach this level too, but they are fused characters.

Super Saiyan IV

This level of a Saiyan is the best. They are no extreme weak points in this stage. Everything speed, strenght, and so on are at its best in the SS4 stage. Goku is the only Saiyan really to reach it Vegeta needs Bulmas creation the Brute Wave to become this. Gogeta, Vegeta and Goku fused, reaches this level too when they fuse at the SS4 level. To become SS4 you need a tail which lets you transform to Golden Oozaru. Then when the Saiyan is extremely enraged he will transform to this stage. Some sites may not include this stage in a Saiyan stage section because it only happens in GT, which wasn't created by Akiro Toriyama.