This stage was first reached by Goku in movie #2 as he fought Lord Slug. He didn't appear to be a Super Saiyan then, since his hair had a black tinge to it.
Officially, Goku went SSJ1 for the first time in the Freeza Saga, after Freeza killed Krillin.
Physical description: Increased muscle mass, and long blond hair that stands on end. Occasionally accompanied by a yellow wave, or lightning. *Note* A Saiyan is normally born with a tail. As long as he keeps his tail, he can turn into a Were-monkey.
Super Saiyan Level 2
I personally think Goku achieved this stage in movie #3, when fighting Cooler. In movie #9, Gohan was told by Goku to go SSJ2, which he did in order to kill Boujack and the rest of his crew. This level is incredibly more powerful than SSJ1. SSJ1 was used quite frequently in movies #4-9, but SSJ2 had more immediate results. SSJ1 proved inefficient in stopping Boujack's crew, and when Gohan reached SSJ2, he easily overpowered the aliens.
Physical description: Greater muscle mass than SSJ1, and a few more inches of blond hair. Occasionally accompanied by zapping silver lightning.
Super Saiyan Level 3
Goku first achieved this level during a fight with Majin Buu. This is the most powerful level of Super Saiyan. Majin Buu could overpower SSJ3, but this level did destroy Fat Janenba in movie #12. SSJ3 was also used in Dragon Ball GT, but Goku couldn't hold it due to his little body's lack of strength.
Physical Description: Goku's physical appearance is more lean and menacing looking, with about 4 and half feet of flowing blond hair and a larger muscle mass.
*Note* Only Goku and Gotenks have achieved this level. No other Saiyan's have achived it.
Super Saiyan Level 4
This is actually not a SSJ level, seeing that Goku doesn't have blond hair. He does have increased muscle mass though, but I don't believe this is a SSJ level. This is also the only stage with a tail. In SSJ4, Goku is almost god-like, and he can use "Times 10 Kamehame-ha."
Physical description: Although Goku was wished by the Dragon Balls to become a child again, in SSJ4 form, he has his normal adult body with a bit more muscle mass. His upper torso is covered in red fur, his hair is black, and his eyes are red rimmed.
When a Saiyan stares at a full moon, he/she will change into a 20-foot tall, monsterous monkey or grollia called a Were-monkey (Oozaru, in Japanese, translates to "Big Monkey). This is possibly the most powerful level under SSJ3. All Saiyan (excluding Goku and Gohan) will be able to control themselves during this stage (although in Dragonball GT, it is implied that Vegeta couldn't control himself in Oozaru form, but Bedi could.) It doesn't matter which full moon they look at (in Dragonball GT, Goku turned Oozaru by looking at the full Earth. And, Vegeta had the ability to create a pocket-sized full moon when he first arrived on Earth to fight Goku and Gohan that made him turn into the Were-Monkey.)
*Note* A Saiyan must have a tail to be able to turn Oozaru. If the tail is cut off, they revert to normal. If the tail is pulled out by the roots, the Saiyan will not turn Oozaru ever again.
Ougon Oozaru (Golden Were-Monkey)
A level that was achieved by Goku and Bedi-Vegeta in Dragon Ball GT. This is the most powerful form of Were-monkey, demonstrated by Goku in DBGT episodes 38-40, when he swatted Bedi-Vegeta all over Planet Plant (Bedi's recreation of the Tsufuru home planet.) The main physical difference with the Ougon stage, is that the fur is yellow instead of brown.