An Exorcism was a ritual used to expell a demon or human spirit which had possessed a human or dwelling. This type of ritual was extremely dangerous for both the performer and the victim of the possession, sometimes resulting in the death of either individuals.
After the demon Eyghon permanently possessed their friend Randall, Rupert Giles, Ethan Rayne, Deirdre Page, Philip Henry and Thomas Sutcliffe tried to exorcise the demon from Randall's body. However, they only succeeded in killing Randall.
In 1997, when the spirits of James Stanley and Grace Newman unwittingly wroke havoc in Sunnydale High, the Scooby Gang tried to exorcise James Stanley, forcing him to move on from this plane of existance. However, the exorcism was cut short by James' spirit.
In 1998, Laibach of the Abyss, an Old One possesed Drusilla while the Master posseesed Angel. Xander Harris and Rupert Giles managed to exorcise Laibach while Willow Rosenberg managed to exorcise the Master after Buffy weakened him, casting a protection spell on him, Spike and Drusilla so he couldn't posses any of the three of them again.
In 1999, Angel and Doyle attempted to exorcise the ghost of Maude Pearson in order to stop the constant murders of the young women that lived in Pearson's former apartment. However, Mrs. Pearson's ghost was strong enough to resist the kind of ritual Angel attempted to perform.
Months later, Angel Investigations came face to face with an Ethros Demon, a powerful possessing demon with a penchant for mass murder, which had taken possession a boy, Ryan Anderson. Angel Investigations sought the help of Father Fredricks, a known exorcist, only to discover that he had been killed in the process of exorcising another Ethros from a young girl. Wesley Wyndam-Pryce failed in expelling the demon, but Angel succeeded in completing the ritual. Afterwards, the demon took on physical form to recover its lost energy and Angel was able to kill it with an axe.
In 2003, some kind of demon invaded the Summers home to stop Dawn Summers from seeing her mother (although it may have been a trick by the First and the demon could have been working for it the whole time) and Dawn performs a spell to exorcise the demon from the house. Dawn succeds in the exorcisim, and given that blood splatters the walls for a second afterwards, may have killed the demon when she exorcised it.