Warning: Wikipedia contains spoilers.
Cordelia was originally introduced as a cheerleader and popular student at Sunnydale High, who naturally looked down on Buffy Summers and her low-on-the-totem-pole friends Willow Rosenberg and Xander Harris. But Cordelia turns to Buffy when she is in danger: "I know you're very strong, and you've got all those weapons... I was kind of hoping you were in a gang." Soon Cordelia learns Buffy's true calling and becomes a member of the Scooby Gang, with a corresponding fall in her social status.
In the third season, Cordelia falls briefly in love with Xander, terminating the relationship when she finds him kissing Willow. Shortly after, Xander discovers that Cordelia's family has lost all their money because "Daddy made a little mistake on his taxes - for the last twelve years." Cordelia was a regular character on Buffy for the first three seasons.
Unable to go to college after graduation, Cordelia moves to Los Angeles to pursue her dream of becoming an actress. Instead, she runs into a producer who is also a vampire, is rescued by Angel, and becomes a regular on the spinoff series, Angel. Angel, Cordelia, and the half-demon Doyle unite to form Angel Investigations, dedicated to helping people in trouble who are usually located by visions sent to Doyle by the mysterious and semi-divine Powers That Be. Cordelia's budding romance with Doyle is ended by Doyle's death, but Doyle leaves her his visions as a dangerous gift.
The visions give Cordelia a powerful ability to help others, but are intended only for demons, since humans are too weak to face them. Increasingly, Cordelia finds herself suffering physical damage from the visions which she struggles to hide from her friends. Ultimately, Cordelia refuses an offer from the Powers That Be to trade the visions and her life fighting evil for her original goal of being a successful actress, choosing instead to become herself part demon. As a demon, Cordelia can receive the visions uninjured, but what other consequences the change will have for her remain to be seen. In the fourth season of Angel, Cordelia returned to Earth as an amnesiac, and then slept with Connor's son, becoming pregnant. It was eventually revealed that her body was possessed by a powerful entity named Jasmine, who used Cordelia to create her own separate body. Once Cordelia gave birth, Jasmine was transferred into the fetus, leaving Cordelia in a coma. Much of this arc was due to Charisma Carpenter's pregnancy. For the final five episodes of Season Four Cordelia was only seen as a comatose body, and after Angel joined the evil law firm Wolfram and Hart, Cordelia was transferred there in an effort to revive her. Since Charisma Carpenter will not be a main cast member for the fifth season, it remains to be seen whether or not Cordelia will be revived.
Like all continuing characters in the Buffy universe Cordelia is evolving: in her case changing from an egocentric fashion plate to a person whose life is increasingly centered in service to others. An interesting aspect to Cordelia's experience is that the changes have come to a substantial degree against her will. The Cordelia we met in the first episodes of Buffy certainly felt no burning desire to become a better or nobler person. The sacrifices she had to make, enduring significant rejection and even mockery from her original friends as she grew beyond their shallowness, were a powerful message for the younger viewers the show is primarily aimed at. Cordelia has now grown into adulthood, with a solid and grounded confidence replacing the arrogance and vanity of her adolescent self.