Originally named Liam, Angel was sired in 1753 by Darla in order to bring him to The Master's Order of Aurelius. After his rebirth, he killed his family and friends and half of the town before he and Darla moved on. They terrorized everywhere they went until they got to London, where the Order of Aurelius was waiting for them. However, Angel decided that The Order was not for him, and taking Darla with him, they left to have their own life.
For one hundred years they lived together, traveling the world, creating death and destruction where ever they went. in 1860, he sired Drusilla after driving her crazy and killing her family. She sired Spike in 1880. The four of them caused much mayhem and became a very feared group.
In 1898, Darla kidnapped a gypsy girl and gave her to Angel as a gift. The gypsy tribe was very vengeful over her death, and curse was put on Angel by the gypsy tribe. The curse restored his human soul. This made him feel terribly guilty remembering all the people he had killed over the past century and a half. Although he tried to get back in with Darla, Spike and Dru, Darla noticed how he would only ever kill criminals. He left after that, and moved to the United States.
Eighty years of living around the US, he met a demon called Whistler, who said he was sent by the powers and encouraged Angel to try and repay the world for his sins by helping the new slayer, Buffy. Angel did. Together they helped others, and fought demons, they even killed Darla and The Master together. After spending a lot of time with her he soon fell in love. On the night of Buffy's 17th birthday, they had sex for the first time. Little did Angel know that the curse the gypsies made had a loophole. It would be broken if ever Angel felt true happiness. On that night, the curse was broken, and his soul was lifted. Angel became evil again, joined up with Spike and Drusilla and this time was even more ruthless. He killed friends of Buffy, and he tormented her for months. Angel had a plan to use the demon Acathla to bring hell to earth by opening up a portal, much the same way The Master tried in 1937. Buffy tried to stop him, and the two fought each other. Meanwhile, Willow worked on recasting the curse, and returning Angel's soul. In the end, Willow succeeded, but it was too late, the door to hell had been opened, and only Angel's blood could seal it. Buffy was forced to stab Angel and send him into hell in order to save the world.
Four months later, Angel was brought back from hell. Time moves slower in hell however, and what was 4 months on earth, was hundreds of years of torture in hell. It took a long time for Angel to recover, but when he did he resumed his relationship with Buffy. It didn't last however, as he realized that he was not normal, and he could never give her a normal life. Angel left Sunnydale to go to Los Angeles where he would try to repay the world for his sins by helping others in need.
So much has happened to Angel in LA, he's set up a detective agency, he's met new friends and found old ones, and he's made plenty of new and old (and resurrected) enemies. He's had a child, and he's even found out about a prophecy that says he'll be turned human after saving the world one last time.
Angel's story is far from over, and that's why I haven't gone into more detail about his LA adventures yet.
Angel is the most seen vampire on both of the shows, and it's him we learn a lot about vampires from.
As an evil vampire, he was called Angelus. When he got his soul back, he decided to go by Angel to separate himself, although Angel was still a name he'd be called by the other vampires who knew him (kind of like friends of a David might call him "Dave.") Like many other vampires sired for The Master, Angelus got the name based on his behavior of how he kills. In this case it was because his young sister thought she was seeing an Angel of her dead brother, and so she invited him in. He of course killed her, and the rest of the family, for her mistake. The name continued to be used because of his Angelic face used to trick people into thinking he's a normal guy.
Something of interest is his relationship with The Master. In early Buffy episodes The Master said things like how he missed Angel and how Angel would have ruled by his side one day if Angel had not regained his soul. Although this may have been some history that was changed later on by the show's writers, I still think this fits quite well with The Master's way of thinking. If you look at their introduction between the two of them, you see Angel totally disrespect The Master, and then have him leave with The Master's favorite sired. The Master does nothing about this, he doesn't have them killed or hunted down and punished. The Master obviously was impressed by Angel, liked his attitude and figured he and Darla would eventually come back. He might have thought since both were sired to be part of the order, after a few hundred years, they probably would have longed to be with other vampires like themselves. This did come true in Darla's case at least, as she did return when Angel left her..
Angel has lived longer then the average vampire seen on the shows. He is very strong, and quite resistant to injury of all sorts, he's survived periods of being on fire, and even had wood a few inches from his heart and lived to tell about it. It's hard to tell if this is normal in a vampire his age because there haven't been many examples of vampires who reached his age, but it's probably just partly due to age, and partly due to naturally being a strong vampire. He also seemed to gain some additional strength after drinking slayers blood, although this might have just been temporary.
Angel also has a human soul, which is what makes him feel guilty for the crimes he has done. See the related link for more on that.