Background; Who are the 'Saiyans'?
The Saiyans are a peaceful race who live alone on a remote planet, doing no harm to anybody. WRONG!! A fierce race of warriors, (Goku and his family are the acceptation) the Saiyans scoured the Universe, fighting, destroying planets, and proving their Physical superiority over and over again. Their strength was further increased once every eight years, when the moon was full on their home planet Vegeta (or what ever planet they happened to be on at the time). Under the light of the full moon, the Saiyans transformed into huge monkey like animals. With this huge increase in strength and size, and the ability to fire blasts of energy from their mouths, the Saiyans were practically indestructible. They all have a natural ability and desire to fight and are very strong (compared to us mere humans!). Raditz was about as weak as they came and when he arrived on earth he was more powerful than the Z fighters had ever imagined anyone could be.
Where do they come from?
Their home planet is or was the planet 'Vegeta'. This planet was ruled by Vegeta's (or 'Prince Vegeta' to give him his full title) father, guess who, King Vegeta (a whole lot a Vegetas).
So how if Vegeta is their home planet, how come Goku is on Earth?
Saiyans are tested at birth to see how strong they are or what their potential strength is. The weak children are sent to distant planets which don't have much chance of destroying them. These children then grow up, destroy the people who inhabit the planet, ready for it's sale by the Saiyans. This is how Goku was sent to earth as a baby, not as an angel from heaven to save it, but to destroy it.
What's with the monkey thing?
All saiyans have a monkey like tail. For a long time, their tail was a weakness because it was very painful when grabbed, tugged or pulled. This caused problems in battle, until they (meaning Vegeta and Nappa) evolved from it. It is the tale that enables them to go Oozaru (giant monkey). If the tail off a full blooded Saiyan is removed it does not grow back (where as the tail of a part saiyan like Gohan does).
What was that about a purebred or full blooded Saiyan?
Theoretically, Full blooded Saiyans are stronger than half Saiyans like Gohan or Trunks, or quarter Saiyans like Pan. If only one parent is a Saiyan, for example, it is still possible for the child to have amazing powers and even go Super Saiyan, they just (again, in theory) are not as strong as the pure Saiyans. The tail of a purebred Saiyan does not grow back once it has been removed (unless the Saiyan reaches SSJ level 4-which until Goku comes along is not that likely). The hair of a full blooded Saiyan does not change from the day they are born (in other words Dave', they don't use a lot of hair gel!!) Of course this is unless the Saiyan goes SSJ, but after that their hair reverts back to normal. A Saiyan also becomes much much stronger after recovering from a serious injury or battle, the harder the recovery, the stronger the person is at the end of it. Back to the subject of purebred Saiyans, they age much slower than normal human beings, and they need to eat a LOT more food ( I think it's something like 3,000,000 calories a day) to keep their strength up and to support their tremendous power.
Super Saiyan? SSJ? Wha'??
After reaching a certain level of power, a Saiyan goes Super Saiyan (or SSJ). The transformation needs a lot of power, but it is USUALLY triggered by emotions such as anger (Goku was the first Saiyan in 1000 years to go SSJ after the death of his best friend Krillen). A Saiyan can only go SSJ if they have a pure heart (see why it was so difficult for Vegeta?) There are different levels of Super Saiya-jin, see Super Saiyan Levels.
Who are the Saiyans we see in DBZ?
Goku was classed as a 3rd class Saiyan at birth and sent to earth because it was a weak planet and he could grow up and easily destroy it. He is the first Saiyan to reach SSJ in over 1000 years, and the first to go SSJ level 3, Ultra super Saiyan and SSJ level 4. See Goku profile.
Vegeta was the prince of the planet Vegeta and heir to the throne. When the planet was destroyed by Frieza, he was one of only 4 Saiyans to survive because he was being held captive on Frieza's ship. He was/is a Saiyan elite, or he would have been on his home planet. See Vegeta profile.
Raditz was Goku's brother, he was at first much stronger than Goku (it took Goku and Piccolo to beat Raditz and cost Goku his life). However, he was a second class warrior.
Nappa was Vegeta's (ahem) 'partner'. He was strong (for that tome in the series) but he was destroyed by Vegeta after failing to even harm Goku in their battle, he was a first class warrior (I think)
King Vegeta was Vegeta's father and the King of the Saiyans. He was destroyed by Frieza when he destroyed the planet.
Brolli was the Saiyan of legend and the first ever Super Saiyan. This Saiyan could go SSJ, but he could only sustain his power in Oozaru form.
Bardok is or was Goku's natural father. We don't actually see him, but we hear from Vegeta that he was an average fighter but a brilliant scientist-the total opposite of Goku. He was also destroyed when Vegeta was blown up.
Gohan Is Goku's son and is half Saiyan, yet he is the first to reach SSJ2, which shows that they can be almost if not as powerful as purebred Saiyans. See Gohan profile.
Trunks is not only half Saiyan, but because he is Vegeta's son, he carries the royal bloodline which has the potential to go Super Saiyan.
Bra is Vegeta and Bulma's 2nd child. She is half Saiyan, but she spends most of her time at home with her mother during battles.
Goten again is half Saiyan like his brother Gohan. He reaches SSJ at age 7, but he never seems to become as powerful and as talented as his father and brother and takes fighting a lot less seriously.
Pan is Gohan's daughter which makes her a quarter Saiyan.