This version is a beefed-up form of a normal super saiyan, who can't quite reach level 2 (ssj2). Muscle bulk increases, but speed decreases during this form. Trunks and Brolli make it to this form, but Vegeta can't quite make it all the way. To reach this form, the Saiyan must have a very high power level, but anger has nothing to do with obtaining it.
Super Saiyan 2
This stage is easily more powerful than normal Super Saiyan form. In this form, hair gets a little longer and muscles bulge more. Only three of the Saiyans reach this stage: Gohan, during the cell games, and Goku and Vegeta both reach it when they are fighting each other and Vegeta is charmed into Majin Vegeta.
Super Saiyan 3
This form is by far the strongest normal Super Saiyan stage. It's one of the most odd looking stages, with hair down to the feet, eyebrow ridges pushed out and eyebrows completely gone. All this physical deformity equals an immense source of power, which sves lives in a pinch. Goku and the fused warrior Gotenks are the only Saiyans to reach this form.
Super Saiyan 4
This form can only be seen in the Dragon Ball GT series. It is not a true Super Saiyan(along the DBZ lines, that is), but it's a mix between Oozaru and human. In this form, warriors keep their hair color, grow red fur across some of ther body( except SSJ4 Gogeta, who had brown fur and red hair) and wear some funky-looking pants. Some believe this is the true form of the Super Saiyan, because the Golden Oozaru can be reached in this form.
His form isn't seen much after the early episodes in DBZ, but they do ocasionally pop up in GT. Oozaru form is when a Saiyan with a tail catches a glimpse of a full moon or an energy level that can mimic it. When they do, they grow fur all over their body, get red eyes and grow to immense porportions. Their power grows 10 times in this form.