Man, which games do I consider classics, and why? Let's see:
Mega Man 2 and 3 (self-explanitory, I enjoyed both games). The soundtrack and gameplay of both games drew me in right away, along with the difficulty. Sure 3 doesn't seem too difficult if you know what you're doing, but once you get to the Doc Robot stages and the Wily stages, the game doesn't let up. MM2 averted the "second one is the oddball in the series" trend that was popular with some games at the time. Of course, that's why I enjoyed both games.
Legend of Zelda: A Link to the Past. I may have played this game later than some people, but it's one of my favourite Zelda games. Once you get to the Dark World, that's when the quotient jumps tenfold. It's still one of my favourite games of all time.
Super Mario Bros. 3 and Super Mario World: I can't decide on which game but there's so much to like about both games. The difficulty stops pulling punches starting halfway in World 5 in SMB3, then gets more challenging for the rest of the game. SMW had a lot of innovations that were almost unheard of for a game 24 years ago.
Super Metroid: Another game I didn't discover until a decade ago, and it was worth the wait. Revisiting old foes like Kraid and Ridley in 16-bit was exciting, but what was more exciting? How the player can break the hell out of some intended sequences, much more than the original Metroid. Do you want to get the Wave Beam early? Yep, that's possible in this game.
Grand Theft Auto: Vice City and San Andreas. Vice City brought me into the series. There's a lot more than just running pedestrians and fighting police officers that got me into the series. One of the reasons is that I enjoyed the characters in both games. In Vice City, you had nominal hero Tommy Vercetti, who went from a nobody out of prison, to essentially taking over an entire city by force. San Andreas had Carl Johnson, who may be a divisive character, but he's involved in some of the best missions in the 3D trilogy, especially the last storyline mission.