Ooh... I like this idea. Though it might be hard to narrow it down to just ten games...
In no particular order...
The Legend of Zelda: A Link to the Past(SNES)
The first Zelda game I really played. I remember coming across it while visiting my cousin and just running around and playing with the items. I was already hooked. A year or two later my grandmother bought it for me and we both fell in love with it.
Starfox(SNES)
It was such a unique game when I played it... The graphics were cool (and revolutionary at the time) and the gameplay was just a blast. It was a difficult game too. I only actually beat it on the level2 and level3 paths for the first time about a year ago. Hehe, I used to remember how to get into the Black Hole, but I lost said knowledge years ago and have never managed to find that blasted face meteor ever since. I saw this one for the first time at my cousin's too, infact it was the same trip I discovered Zelda. Sometime later I saw it in a pawn shop and grabbed it up. (Or did I buy it in the pawn shop first? Gosh... now I'm not so sure...)
Rex Nebular and the Cosmic Gender Bender
...I think I'll just keep this one for myself.
Perfect Dark & Goldeneye007
I list these two in the same place because they were both so similar. Goldeneye was an outstanding FPS when it came out, and it still is. There was nothing else quite like it. Perfect Dark was just like Goldeneye, but expanded upon with even more great content. My friends and I played both games for years...
Final Fantasy VII, VIII, & IX
What can I say? They were the first RPG's I ever played and from the moment I first saw each one I was hooked and new I'd have to have them. I played them all to death and got very good at them. I think they all had great stories, characters, and graphics (for their respective times). Each one was just a blast to play and they had great music too.
Resident Evil 2
Mmm... once again, I saw a friend playing it and I was instantly hooked. It was the first Resident Evil game I purchased and beat myself. I love the setting, the weapons, the characters, the story... the monsters... everything about it. It along with Final Fantasy and the other RE games are what finally convinced me to stop being a Nintendo fanboy and get a Playstation/2.
Resident Evil: Director's Cut & Resident Evil 3: Nemesis
These two are grouped separately from RE2 because they were just so damn hard. RE3 on Hard mode took lots of planning and strategy, and lots of adrenaline to dodge enemies. It was a great accomplishment to finish that game on hard. The same with RE: Director's Cut (and the original RE); Hunters in the original were just so damn dangerous and menacing. After I first encountered them in the mansion I was decaptitated a few times and had to put the game down for months. None of the other games have ever had an enemy that was quite so heart-pounding to face off against. You didn't just fight a few of them either... No, you had dozens and killing each one that you were forced to cross paths with was an accomplishment.
The Resident Evil Remake
It was one of the only actual remakes in any media I can think of that truly just remade the original without making it unrecognizable. It was a bit easier (the Hunters were toned down) but it contained plenty of the old fun, with enough things switched around to make you get lost all over again as well as scare the living bleep out of you. It had great graphics too, as well as voice-acting. Though the cheesy voice acting of the original had it's own charm.

...and you can't forget the killer live-action FMV's.
The Legend of Zelda: Link's Awakening
Well I've talked about it elsewhere on the forum already, but to summarize: it was just a great game all around. A great Zelda game. It didn't feel like a gameboy game. It felt just as solid as A Link to the Past. It had, in my opinion, the best story and setting of any of the Zelda games. The crazyness of Koholint Island was prevelant, but not overbearing or overdone. It took itself seriously, but was light hearted at the same time and provided many challenges with so many secrets to discover.
Command & Conquer, Command & Conquer: Tiberian Sun: Firestorm
Like so many of the other series listed here; when I first saw C&C on a friend's IBM I loved it. To date, no game has quite captured the feel of the very first C&C. It was radical and strange, but believable at the same time. C&C kicked off the RTS genre with it's unique yet familar world, awesome units, and unforgetable storyline and arch villian: Kane himself.
"The rivers will flow with the blood of those who oppose us." --Kane, Nod
Tiberian Sun itself wasn't quite as good as C&C due, I think, to the 3rd person perspective FMV's with actors playing the role as the actual commander. However Firestorm reintroduced the storytelling in a form as though the generals were actually talking to you, and it brought in some great additions to the main game. (units, maps, and such) It was a hell of a challenge to beat too. (Oh, and the music in the games rocked too)
MegaMan X
One of my all time favorite Mega Man games. I first fell in love with the series when I was very little (3 or 4) when I'd go over to my friends and he'd play Mega Man 3 on the SNES. Years later I finally got Mega Man X and beat it on my own. The same with X2, X3, X4, and X5. I loved each one, though X5 felt a bit unfinished. It was still a fitting conclusion to the X series. (according to the original producer X5 was the end)
Oh darn... just one spot left... what to put? Road Rash 64? Turok: Dinosaur Hunter? Duke Nukem 3D/64? Starcraft? Aliens Vs Predator 2? Ah... I know... how could I forget.
Arcanum, Of Steamworks and Magick Obscura
I still play this one reguarly to this day. It is a true classic and a very unique RPG. The concept of Magick vs Science was quite original when it came out. Sadly Arcanum was heavily pirated and the developer who made it (Troika) is no longer around, but it's a great game none the less (as it was voted in several magazines when it was released). The setting, especially, is a fresh twist on RPG's. Instead of a sci-fi world, or medieval landscape we find ourselves in a land reminiscent of Victorian England in the late 1880's. There's too much to say here, really, however Arcanum will always be one of my favorite games of all time, right up there with Zelda.
Edit: Just let it be known there are many games that deserve to be on this list. I already mentioned a few, but Mario Kart/64, the Bloody Roar series, Super Smash Brothers/Melee, the Sonic the Headgehog series, Rogue Squadron 1&2, Red Faction, the Half-Life series... or Grand Theft Auto games? As I said, there are just too many to list and not enough space. Suffice it to say there are lot of great games out there that will forever be classics to me.