Boy, it's not easy.
I didn't spend a lot of time on picking the episodes, because I was afraid that I would just get frozen trying to choose between them all. So I went mostly with impulse picks by scanning the episode titles. I'm sure if I thought about each show in detail, I'd never be able to decide on just 10.
1) Becoming - I had to cheat and count both episodes as one, in order to make only ten. The story that convinced me that Buffy was the best on TV. When she sends Angel to hell and the voice of Sarah McLaughlin begins that sad song, even though I am a mostly unsentimental man and over the age of 40, I wept - no sobbing, just couldn't keep my eyes from leaking.
2) The Gift - The second best finale. The best one-on-one fight, even if Xander did lend a heavy hand at one point. Each of the characters had something unique and vital to add to the episode. And yes, the ending made me cry - again! The sure sign of a winner story.
3) The Body - I can't believe no one has mentioned this one yet. I just regret that they had to add that lame vampire at the end. It was a perfect episode without any monsters. Most shows have a character die and the very next scene is the funeral and then everyone moves on. This story took us minute-by-painful-minute through the death of loved one. No TV drama that I am aware of has ever dealt so unflinchingly with this subject. Without a doubt, it should have won the Emmy that year.
4) Anne - The perfect follow-up to Becoming. Despite her best efforts, Buffy is forced to face the fact that she is a hero. Lily was a great character as well. The best Buffy vs. a whole gang of bad guys fight. When she was up on that raised platform kicking the ass of all comers, it was a rush. It also gets the nod as one of my favorite Buffy-as-Bruce-Willis lines, "Anyone not having fun here, follow me."
5) Graduation, Pt. I - I could have cheated and added Pt. II, but that would have cheapened Pt. I, which was far better. The definitive Buffy vs. Faith showdown. The dismissal of Wesley combined with Buffy's "graduation" from the Watchers. Buffy's and Faith's mystical connection through dreams. Anya professing her confounding love for Xander and becoming a member of the team. But mostly - Buffy vs. Faith.
6) The Wish - Speaking of Anya. The dark, morbid alternate universe. Cordy getting what she deserves for creating that hell world. Evil Willow. Puppy Angel. And Giles saves the world from Anya and the Master, which he doesn't get enough credit for, because no one remembers it.
7) Innocence - The introduction of Angelus; Jenny's secrets revealed; The Judge; Angelus-Spike-Dru together again for the first time; Willow and Oz sitting in the van - "See, in my fantasy when I'm kissing you, you're kissing me. It's okay. I can wait."; and, of course, the rocket launcher - "What does that do?"
8) Once More, With Feeling - a gimmick show, but a gimmick that worked. They could have done all the character and story development without singing, but they were also talented enough to pull off the challenge. I have a soundtrack CD I created of the songs and I still play it at work sometimes and sing-along.
9) Faith, Hope & Trick - Two great characters introduced in one show. Sadly, Trick did not live up to his potential, but he had great beginnings. Faith as the anti-Buffy. Faith getting to know the gang at The Bronze is a laugh from start to finish. Faith seeming to thoroughly enjoy being a Slayer. Joss doesn't seem to keep edge-y characters around for long, but it was obvious from the start that Faith was a keeper.
10) Hush - Another gimmick show, but another one that worked. The briefing by Giles without words is a classic comedy bit. Riley meeting Buffy as The Vampire Slayer for the first time. And the Gentlemen were great bad guys. One of the most creepy moments of all time was contemplating having your heart cut out and your screams can't even be heard. - OH! I forgot and so I'm adding this later - I can't believe I didn't remember that this is the beginnings of Willow and Tara's relationship. For that reason alone it belongs in the top ten.