Capcom vs. SNK 2: The Groove system is one of the coolest dynamics to tailor the game to your playstyle. Is it balanced? Hell no, but I love this game casually.
The King of Fighters 2002: KOF fans will tell you either 98 or 02 were the absolute pinnacle. I side with 02 because it has Kula in it. Also note that 98 and 02 both have updated rereleases with an extended roster and rebalancing, but those are Windows-only.
Puzzle Bobble 1/3: You've probably played some flash game clone of this. IMO I think 1 was best for its simplicity, I'm not as fond of the garbage patterns introduced in later titles in an effort to give characters some asymmetry. But PB1 does not have AI opponents, singleplayer is only the stage clear mode, so if you don't have a human to play with try PB3 for the next best thing.
Soldam: The singleplayer modes are nothing to write home about, but it has one of the most unique versus modes I've seen in a puzzler. Shared piece queues are normally horrifying, but Soldam makes it work by giving P1 the objective to match red while P2 matches blue. So if you want to snipe pieces that are desirable to your opponent, that means taking pieces undesirable to yourself. Garbage is also based on how you clear lines, so crafting maximally disruptive garbage gets interesting. The catch, unfortunately, is that there is no AI. But if you can play this game with a human, do check it out.
Tetris: The Grand Master 1/2/3: The only good Tetris, do not @ me. Start with TGM2's Novice Mode, then once you can clear that go back to TGM1.
Twinkle Star Sprites: A versus shmup with a very unique format. Chaining enemies on your screen sends attacks to your opponent's screen. Hard to really explain, just give this a spin and feel it out for yourself. There are a lot of moving parts, screenwatching is vital, and feels like I've barely scratched the surface of the game's depth.
Vampire Savior: Aka Darkstalkers 3. This game is fast as hell and it's a blast. Like with any classic fighter, good luck keeping up with FightCade folks who really know what they're doing, but I love it casually.
Waku Waku 7: This game's mechanics are honestly borderline kusoge, you can't even cancel normals into specials. But I love the design and atmosphere so much. Tesse is really fun to play even in spite of the system mechanics.
NES:
Fire 'n Ice: A very rad little puzzle game.
Mike Tyson's Punch-Out!!: Just an absolute blast. I won't bother listing them seperately but also check out Super and Wii. Super's kinda the black sheep of the series, but it's still a good game, just not as good. Wii is an absolutely top-notch successor and I'm sad it didn't get any more sequels after that. The two arcade predecessors are honestly forgettable.
SNES:
Chrono Trigger: I am hesitant to recommend most JRPGs from this era if you did not grow up on them, because many of them haven't aged so gracefully. Chrono Trigger is the exception, this game is a fine wine. You may want to check out one of the rereleases though, or at least a retranslation patch, because the original translation was made on a rushed deadline and bound by heavy technical limitations.
Earthbound: A bit more of a slow burn in comparison to CT, but this game is carried by incredible writing. It's also required reading before playing Mother 3 next.
Kirby Super Star: Definitely the peak of the series, giving every copy power an entire moveset is a blast. Has an updated rerelease on DS with added extras, I do highly recommend this version, but DS can be awkward to emulate so SNES is fine.
Wario's Woods: The NES version is more well known since it was the system's last first-party title, and for whatever reason it's the only version Nintendo ever rereleases. But the SNES version is a notable upgrade, biggest thing it has is AI to play versus mode against. Versus mode is wild as hell, so if you've never seen it please check out the SNES version.
N64:
Dr. Mario 64: Best version, but can be notoriously difficult to emulate. If you have issues with it, SNES is a good alternative. Don't play NES.
Mario Party 2: Still the best in the series.
Paper Mario: Pure perfection. Many fans will say TTYD was better, and it's certainly a good game too, but I think 64 was peak simply because the pacing is so much better.
Super Mario 64: It's Super Mario 64. You do not need me to tell you that this game is good.