Angel was born in the early 1700s. Although as a human he never left Ireland, once Darla had changed him into Angelus the vampire, he started touring Europe on a evil killing spree where he did some sick, twisted things including driving people completely insane and killing every person they knew (a.k.a Drusilla). Things changed in 1898 when he killed the favorite daughter of a Gyspy Romany tribe.
Gypsies cursed Angel to have a soul and conscience. As a result of the curse Angel has to live with memories of what he did as a soulless demon. With these memories tormenting him, he still lives off blood, but now, unlike every other vampire -- he won't kill to get it.
Angel is quiet, miserable and truly regrets what he has done as Angelus - the soulless vampire that killed everything it came across for over a century. Staying out of sunlight, and not killing anyone has meant Angel has been alone for a long time.
Throughout series one, Angel pops up and either offers Buffy cryptic advice or blatant warnings of impending danger.
To make a long story short, Angel became Buffy's boyfriend, although it must be said that seeing how they only spend their time together trying to save Sunnydale/the world that it isn't quite as good as it is cracked up to be.
Unfortunately, raging hormones got in the way and Buffy and Angel make love. Causing Angel to lose his soul and become Angel again. (Ain't That a kick in the ass!!!)
Angel, now Angelus, starts planning all kinds of evil ways of getting back at people who made him feel human - Buffy and her friends. He also gets this 'great' idea to open a portal to hell.
He spends most of Season 2, torturing Buffy and her friends, this includes killings Mrs. Jenny Calendar (He snapped her neck in two...dirty bastard..).
At the end of season 2, Buffy and Angelus fight, Willow casts a spell that reverts Angelus back to Angel, returning his soul. Unfortunately for all concerned -- it is too late to stop the portal to hell being opened by Angelus. The only way for Buffy to close the gateway to hell is to impale Angel with a sword and then send him to hell.
All is not lost as later in season 3 Angel returns .
He and Buffy share a bunch of 'I love you but I can't be with you' scenes and we feel sorry for them and stuff.
Well, not all of us actually, I'm sick of the whining! I want smoochies! I want passion!! That's why I want Willow and Angel together!
Of course Angel is now getting his own series set in Los Angeles (Isn't THAT convenient!)
But I'm still striving for Willow and Angel to become a couple!