Welcome to the Hellmouth, Mookeylama!
Yes, the destruction of the Hellmouth, Anya's death and Spike's sacrifice, all occured in the series' final episode, "Chosen."
Buffy the Vampire Slayer debuted on the WB on March 10, 1997, and bowed-out on UPN on May 20, 2003, after seven seasons of some of the best television ever produced.
Resources for more info:
Overview:
Wikipedia.org
Episode guides:
BuffyGuide.com
Trivia:
The Buffy and Angel Trivia Guide
Trailers, transcripts, episode summaries, screencaps:
BuffyWorld.com
Episode transcripts:
TwizTv.com
Screencaps:
ScreencapParadise.com
Critical studies:
Slayage.tv
All things Buffy/Angel/Firefly/Serentity-related:
Whedonesque.com
Edit: Since you're watching on FX, here's a link to
FxNetworks.com. Click on Shows>Buffy The Vampire Slayer for broadcast schedules, episode guides, and behind-the-scenes video bites.
The series' premiere episode, "Welcome to the Hellmouth" airs tomorrow, 9/20. The show is a bit tentative as the writers and actors search for their voices but, as The ObsoleteMan stated, below, the series comes into its own, big time, in Season 2 with the episode "Surprise."