Welcome to the Hellmouth (Part 1 of 2): Buffy arrives in Sunnydale and is enrolled at Sunnydale High School after burning down the gym of her old school in LA, because it was full of vampires. The general public, of course, do not know the vampires were there and see Buffy's act of merely one of arson. Buffy is trying to escape her destiny as a Slayer, but the fact that Sunnydale is swarming with vampires is not conducive to this hope. She meets her Watcher, Giles, but is hostile towards him. As the episode ends, Buffy is lying in a coffin, being attacked by a particularly imbecilic looking vampire, while her new friends Xander and Willow are standing in a graveyard, being attacked by another particularly imbecilic looking vampire ...
The Harvest (Part 2 of 2): Can't remember much of this one, because Sky broadcast it absolutely ages ago, and of course it didn't make much sense without Part 1 (which, in their wisdom, Sky did NOT broadcast), whereas BBC put Welcome to the Hellmouth on only this evening, so its events are fairly fresh in my mind. I do remember that it involved Buffy and Xander running around the sewers looking for a particularly imbecilic vampire (come to think of it, they all look particularly imbecilic) called the Master. And as the Master later appears in Prophecy Girl, I'm guessing they didn't find him.
The Witch: This episode seemed to say that witches were predominantly bad, because the baddy in this episode was a witch, who, for some reason, was swapping bodies with her daughter in order to become a cheerleader. Funnily enough, though, when Willow becomes a witch much later, it suddenly seems as though witches aren't quite so bad after all.
Teacher's Pet: A new teacher comes to Sunnydale High after the previous teacher was killed. It therefore comes as no surprise when the new teacher is revealed to be the one who killed the old one so she could get into the school. It comes as slightly more of a surprise, however, when the new teacher also turns out to be a praying mantis who wants to kill Xander.
Never Kill A Boy On The First Date: The rather catchy title is somewhat misleading, as Buffy of course does nothing of the sort. She goes out with somebody called Owen, who, unlike his namesake (that's the owner of the website, by the way - me), turns out to be rather psychotic in that he gets a buzz from nearly being killed when he sneaks along to one of Buffy's slaying sessions. After this, Buffy dumps him.
The Pack: In this perhaps the most ludicrous episode of the first season, Buffy, Willow and Xander go to the zoo. While there, Xander and four other students get possessed by hyenas, and thereafter run amok for a while, generally behaving like hyenas, until the zookeeper returns them to normal by becoming possessed by a hyena himself.
Angel: The rather nice individual called Angel who has been going around liberally distributing leather jackets in Buffy's direction since the beginning of the series reveals that he is in fact a vampire. But he's a nice vampire, so it's okay.
I Robot - You Jane: Willow encounters a boy called Malcolm on the internet who she thinks she falls in love with. Unsurprisingly, however, Malcolm isn't actually a boy at all, he's a demon who was imprisoned in the pages of a book and released when the pages of the book were scanned into a computer. And Willow should have known it all along.
The Puppet Show: At the school talent show, Buffy notices that a number of people are being killed and having various organs removed. This is not good, especially since the Principal, an unpleasant chap named Snyder, believes Buffy has something to do with it. After everyone thinks it's a ventriloquist's dummy that can walk and talk (come on, it's not so improbable that everyone would blame this particular mutation of nature), it turns out to be a demon disguised as a boy who does magic tricks. But Xander (get this, XANDER!) kills him.
Nightmares: A thoroughly unenjoyable episode in which everybody's nightmares come to life. One is tempted to wonder, after seeing that Xander's worst nightmare is a psychotic clown, whether the poor lad is quite all right in the head.
Out of Mind, Out of Sight: An invisible girl is terrorising the pupils at Sunnydale, especially Cordelia, the popular girl who everybody likes and who obviously doesn't like Buffy. It turns out that the invisible girl was made invisible through quantum mechanics - because everyone ignored her, she became invisible! Anyway, the episode is only noteworthy because of a truly great line from Xander. At the point when the gang doesn't know that they're being attacked by an invisible girl, she hits somebody with a baseball bat. When wondering how this could happen, Xander suggests, "Maybe it's a vampire bat." Chortle!
Prophecy Girl: Giles uncovers a prophecy that Buffy will die in combat with the Master after an earthquake, and an earthquake has of course just occurred in Sunnydale. Buffy, although obviously not terribly pleased, embraces her fate and goes to fight the Master, and (shock! horror!) actually dies. However, the prophecy did not state that she couldn't come back to life again, which she does in due course, with a little help from Xander and Angel, a friendly vampire. She then kills the Master.