You obviously haven't been watching too much DragonballZ or GT if you ask that.  I can understand it if you're a big fan of the original "Dragonball".  Nobody ever knew exactly what Goku was until his big brother Radditz came to Earth and told him of his alien race and his true reason for being alive.  And what he told everyone gave a new perspective to the series.

Saiya-jins are from the Planet Vegeta, also known as Planet Plant.  Long ago, apparently while Goku, Radditz, and Nappa were out on a mission and Vegeta had been handed over to Furiza, their planet had been destroyed by a "meteor shower", later revealed to actually have been a huge blast that Furiza had sent.  (Nice guy, huh?)  Hence, only four Saiya-jins remained.  Even the king and his guards had been destroyed in previous battle.  After that, Gohan was born, who was half-Saiya-jin and half Earthling.  That's five.  Then Radditz and Nappa were killed, and that's three.  Then Trunks was born, going back to four again.

Full-blooded Saiya-jins in their normal form have black eyes and black hair, or are bald, as is the case of Nappa.  (Believe it or not, he used to have hair!  Kind of like Recoome's, but black.)  If the Saiya-jin has mixed blood, say half Earthling, he or she may still have the black hair / eyes trait going, but sometimes, as in the case of Trunks, the traits may do something a little out-of-the-ordinary, such as having purple hair and deep... baby blue... eyes........  <<Sorry.  Trailed off there.  :) >>

Full-blooded Saiya-jins, and some half-and-half ones, are born with a tail that makes them able to transform into Oozaru1 form after seeing the full moon².  They can destroy entire cities in moments in this form, and it's reeeeaaallllyyy scary.  If the Saiya-jin is still weak, they cannot control their powers and are capable of killing anyone, even if it's a good friend or family member.  When this weak Oozaru turns back to normal, he or she will not remember anything about what they did.  If the tail is just pulled off, which happened to Gohan twice by Piccolo, then it grows back.  However, if cut, the wound does not heal and therefore the Saiya-jin cannot go into Oozaru form again, even if put into a Rejuvenation Tank.³  When a Saiya-jin is in this form, his/her power increases tenfold.  (That's power level times ten, in laymen's terms.)  Then there's the size factor and the thought of having that huge beast trample over the city, not to mention your house, your car, and <<dare I say it>> you.

Now on to the really tricky stuff:  Super Saiya-jins!  The birth of one of these guys (or girls, but I have yet to see any...  \_/  Evil...) is one in one-thousand.  (I don't get how Viz and Funimation got one every thousand years or one in three thousand years...  That's really dumb.  It doesn't even fit the plotline!  And if you say there's no plot...  There is.  And if you don't believe me, do you know anyone who watches the DB series for the plot?  Thank you very much.)   In order to become a Super Saiya-jin (SSJ), you must have all four of these qualities...
1)  Must have an extremely high power level.
2)  Must be pure of heart, which is either pure good or pure evil.
3)  In order to transform, must have strong emotion over something
4)Must have at least half-Saiya-jin bloodline.

And I believe myself to be one.  What?  You have a problem with it?  Yeah.  Try saying that to my face.  I'm a Saiya-jin, and chances are you're a mere Earthling.  Yeah.  Just try it.

For example, when Goku first went SSJ, he was on Namek.  This was after he trained under 100x gravity and was healed in a Rejuvenation Tank, so it's obvious he has a really high power level.  He is, of course, pure good.  And then Kuririn, his best friend since they were kids, is killed right in front of him by Furiza.

Vegeta was the second to go SSJ.  He had trained at 400x gravity, I believe, and he was mad because he still couldn't go SSJ.  He knew that he could only defeat "Kakarotto" if he was transformed, and he became mad at himself.  "Why can't I do it?  I...  am...  PURE... EVIL!!!"  At that moment, he went SSJ.

Chibi-Trunks first went SSJ when he and his father (Vegeta) were in the gravity training room.  "Okay son.  If you can land a good punch on me, I'll take you to the carnival."  "THE CARNIVAL!?"  Trunks went SSJ.  (Ain't that cute?)  It appears that he was pure good, judging by how he acts.

Not too sure how Gohan does it, or Brolli, but I'll find out sooner or later.  One thing I do know about Brolli, he could only sustain his power when he was in Oozaru form.