While only a few true full-blooded Saiyan (or more properly: Saiyajin) in the "official" Dragonball Z series (namely, Son Goku, Vegeta, Raditz, and Nappa), many others show up in the numerous Dragonball Z movies, including Bardok, Goku's father, in the Bardok TV Special, and as the enemies in several of the movies, including The Tree of Might (AKA "The Mega-Vigorous Battle for Planet Earth"), where the leader of the group trying to destroy the Earth looks exactly like Goku, except his skin is gray (the reason this is so is never explained). Also, there is an enemy who shows up in a couple of the movies named Brolli, who is apparently quite tough (his movies have not been released in the US at the time of this writeup)
The Saiyans were incredible fighters. Not only were they very strong from birth, each time they are heavily injured and then recover, their strength increases dramatically. For some time, they worked for Frieza (whose name means freezer; his brother was named Coola, or cooler), a mysterious and very powerful being who used them as troops to decimate planets, which he would then sell to other races looking for a new home to colonize. But Frieza ended up destroying planet Vegeta because he feared the power of the Super Saiyans, a power which had not been seen in a thousand years. When a Saiyan reaches Super Saiyan, his hair glows yellow, and an aura surrounds their body. They become much faster and stronger. Only a few Saiyans survived the destruction of Vegeta, including Goku. For all the neat things that happen to Frieza on Namek read my writeup in Frieza.
An interesting thing about the Saiyans is that they are all named after vegetables. Vegeta, naturally, just means vegtable (as he is named after the planet that the Saiyans are from). Nappa, his sidekick, is named after cabbage. Raditz, Goku's brother, is a radish. The word bardok, Goku's father's name, refers to a kind of root vegetable native to Japan (you can get it at finer sushi restaurants). And Goku's birth name, Kakarrot, means carrot - he was named Goku (monkey) by an old man who found him after his spaceship crashed on Earth.