Super Saiyan 1:  The Saiyan's hair turns gold and changes the shape of their hair.  They become more powerful than their original state.  Their eyes turn green and their speed increases.  Their strength improves too.  When Goku is in this stage, he cannot do a Genki-Dama (Sprit Bomb) or a Kaioken attack.  If he does, he will become too tired to finish the move.

Super Saiyan 2: This stage was first accomplished by Gohan in the Cell Saga.  This stage looks exactly the same as the Super Saiyan 1 stage, except for the eyebrows.  The strength and speed is much greater than the Super Saiyan 1 stage.

Super Saiyan 3:  Goku has first reached this stage.  The Saiyan's hair grows as long as their body.  The hair is a gold color and their strength and speed improves enormously.  The eyes like the other stages are green.  In this stage, the character can use new moves they cannot do in another other stage.  Only Goku and Gotenks have ever made it to the Super Saiyan 3 stage.  You do not have to go Super Saiyan 2 first to go into the Super Saiyan 3 stage.

Name: Goku
Super Saiyans-jin: In the Freeza Saga, Freeza kills Krillin which angers Goku which triggers him to go Super Saiyans-jin.
Super Saiyans-jin 2: When he is training in the afterlife.
Super Saiyans-jin 3: When he is training in the afterlife.
Super Saiyans-jin 4: In DBGT in the Baby Saga, Goku gets his tail pulled out; he turns Oozaru then Ougon Oozaru (Golden Giant Monkey) form; when he sees his granddaughter(Pan) crying, it triggers him to go Super Saiyans-jin 4.

 
Name: Vegeta
Super Saiyans-jin: When he was training in the Gravity Room, he became very strong but not strong enough to become Super Saiyans-jin; he gets very angry at himself which triggers him to go Super Saiyans-jin.
Super Saiyans-jin 2: He reaches this stage while being Majin Vegeta.
Super Saiyans-jin 3: Never Did.
Super Saiyans-jin 4: He cheated for this stage; Bulma made a Brute Ray which made Vegeta go Super Saiyans-jin 4 therefore he cheated.

 
Name: Future Trunks
Super Saiyans-jin: In the T.V Special "Resist Dispair!! The Ultimate Surviving Fighters... Gohan and Trunks", when Gohan dies Trunks gets mad and he becomes Super Saiyans-jin.
Super Saiyans-jin 2: Never Did.
Super Saiyans-jin 3: Never Did.
Super Saiyans-jin 4: Never Did

 
Name: Gohan
Super Saiyans-jin: When he was training in the Spirit Room, he got enraged that he couldn't help his friends, which triggerd him to go Super Saiyans-jin.
Super Saiyans-jin 2: When Cell killed Android #16.
Super Saiyans-jin 3: Never Did.
Super Saiyans-jin 4: Never Did.

 
Name: Chibi-Trunks
Super Saiyans-jin: While training, he did not need extreme rage to trigger his SSJ form. Vegeta was surprised by this.
Super Saiyans-jin 2: Never Did
Super Saiyans-jin 3: Never Did.
Super Saiyans-jin 4: never Did.

 
Name: Goten
Super Saiyans-jin: While training, like Chibi-Trunks he also did not need extreme rage to trigger his SSJ form. Gohan was surprised by this.
Super Saiyans-jin 2: Never Did.
Super Saiyans-jin 3: Never Did.
Super Saiyans-jin 4: Never Did.