Carlisle Cullen is an incredibly handsome, conscientious, morally-sound doctor. As patriarch of the Cullen clan, Carlisle started the whole "vegetarian" (no-human) diet. Carlisle is a vampire with a conscience.
As a human in the 17th century, Carlisle was the son of an ardently anti-"evil-being" (witches, vampires, werewolves, etc.) pastor. Carlisle followed in his father's footsteps and hunted and killed these "monsters." While chasing down a vampire, Carlisle was bitten. After hiding in a pile of rotting potatoes while he underwent his painful transformation into a vampire, Carlisle realized what had happened to him. Horrified at what he had become, he tried for years to kill himself, but nothing seemed to harm him. Eventually, Carlisle reached a sort of peace with himself when he decided to hunt only animals. He gained enough self-control that he could even handle being around human blood, and eventually became a doctor. An outsider among vampires because of his diet, Carlisle, like the Pilgrims that settled in New England, traveled to America to start a new life.
Just as Carlisle has rules about his diet, he also has a set of rules regarding creating new vampires: he will only transform people who are about to die and have no other option. When the human Edward was about to die of Spanish influenza, Carlisle transformed him. He also saved his wife Esme, Rosalie, and Emmett. One of Carlisle's most important characteristics is his compassion, and when he transforms someone into a vampire, he's sure to look after him or her.
We're not sure if Stephenie Meyer had this in mind when designing Carlisle's character, but we find it interesting to consider is how Dr. Carlisle Cullen and Dr. Victor Frankenstein from Mary Shelley's Frankenstein stack up next to each other. Both men create "monsters," but the primary difference between them lies in how they treat their creations. Dr. Frankenstein is horrified by the monster he's made and abandons it. Carlisle, on the other hand, invites his creations into his home and treats them as family. In fact, we find out from Edward that to abandon a vampire you've created is considered unethical, at least by the Cullens. Alice was alone when she woke up a vampire, and the Cullens are amazed that she didn't become a "total savage" (14.44) as a result. Interestingly, in contrast, Dr. Frankenstein did abandon his monster, and the monster did become a "total savage."
As you might expect, Dr. Cullen is a family man, and he's open to expanding his family. When James, Victoria, and Laurent show up in the middle of the Cullen baseball game, Carlisle introduces his family, and includes Bella among them.