Continuity mistake: During season 7 we are told several times that the First cannot take corporeal form, ie. it cannot touch things. However, in its first appearance in the season 3 episode 'Amends', when it manifests as Jenny Calender and runs its fingers through Angel's hair you can see his hair moving underneath her fingers. It wouldn't be able to do this if it weren't corporeal.
Continuity mistake: When Buffy is fighting the vampires at the end of the episode, she grabs a tennis racket and smashes the brunette vampire over the head. She falls over the back of the couch and we hear the thud of her landing, however in the wide shot of the couch the vampire has just started to go over the back.
Other mistake: Joyce tells Cordelia to grab some bandages for Xander from the bathroom upstairs. Joyce and Xander are in the kitchen and there is a small corridor that leads to the stairs, but Cordelia goes around the other way leading to the living room. Fair enough, Cordelia made a mistake. She isn't that bright. But she returns with the bandages coming back from the living room. Why come downstairs and then go all the way around through the living room when she could have just walked down the small corridor?
Continuity mistake: When Buffy has told the Scoobies her plan about going down into the Hellmouth, Willow starts talking about how it's beyond anything she's ever done. As she says this she is looking in Buffy's direction, but as she continues and the camera cuts away, her reflection can be seen in a nearby mirror - she is suddenly looking straight ahead of her and then turns to look in Giles' direction. She is then suddenly looking back in Buffy's direction as the camera cuts back to her.
Visible crew/equipment: When Dawn and the Potentials try to get past one of the police officers at The Bronze, Kennedy says that he'll have to shoot them all to stop them. As he responds with "It doesn't really bother me", the shadow of a boom mic can be seen on the wall behind him in the top right hand corner of the screen.
Other mistake: When The First (disguised as Eve) tells the Turok-Han to kill everybody except Buffy, it runs past her towards the exit of the crypt. The First then says, "Well, alone again", turns to face Spike and transforms into Buffy. In this shot, the Turok-Han can be seen still leaving the crypt, even though it had plenty of time to reach the exit and was running towards it at full speed.
Continuity mistake: When the Buffybot is at the parent teacher day and she is looking at the student model of a Utopian city, the hover cars are moving around the model on a wire. But when the shot changes, the cars aren't moving. Then in the next shot, they're moving again.
Revealing mistake: When Gwendolyn Post fights Angel in the mansion, she breaks a long-handled shovel over her knee to create a stake. If you watch in slow motion, the handle of the shovel is pre-broken (she simply holds the two pieces together tightly), and the pieces separate just before they actually make contact with her knee.