Angel is the name of a vampire also known as Angelus and Liam. Originally a rather rakish young man in the 18th century, he was turned into a vampire by Darla the vampiric 'child' of Orlock-esque Master. For many decades after this he was an exceptionally cruel member of the living dead, until he slaughtered a gypsy girl. Her tribe cursed him in revenge by returning to him his soul, whereupon Angel went nearly mad with grief. Not until the latter half of the 20th century did he find a purpose again, after falling in love with Buffy Summers. Eventually he established a supernatural detective agency in Los Angeles, California.
His story has been chronicled in two television series, several novels and a comic book, all under the supervision of Joss Whedon.
Liam was his mortal name. Angelus was his name as a more-or-less ordinary vampire, based upon his handsome face. Angel is the name he chose for himself as an ensouled vampire.
Charles Loridans (see below) included Angel in his timeline "Children of the Night."