Amazing games you wont forget

Mindjack :P

Asura´s Warth (Xbox 360)

I have just recently played Mad Father that really had an impact on me.
Its a free indie horror adventure game (RPG style), and I can say that it is awesome,
albeit short (you can finish it in ~3hrs).
Site for download

But this developer has also several similar games, that I definitely plan to play when I find more time.

The Longest Journey, Alan Wake

1.Fahrenheit(Indigo Prophecy)
2.Half-Life Franchise
3.GTA: Vice City
And some others...

Well, if we're talking old school... Suikoden II for the PS1 is my favorite game of all time.

to the moon

The Portal series, The Legend of Zelda Series and I wanna be the guy. It took me 3 months to beat IWBTG

PC: Half life 1 / Thief / SWAT 4 / Sin / C&C: Red Alert / Diablo / Op Flashpoint: Cold War Crisis / GTA: VC

SNES: Super Mario Kart / Flashback / Super Mario 2 / Super Strike Gunner / Starwing / Legend of Zelda: A Link to the Past

Minecraft more recently

games I haven't forgotten from my childhood include, Dragon Warrior I (dragon quest), Faxanadu, Heroes of MM 2, Zelda OoT.

Quake2, Age of Empires 2 and many more
MANY

Every zelda game, Final Fantasy 9 (PS1) , Legend of Dragoon (PS1), Pokemon (1st ,2nd and 3rd gen), and Castlevania : Symphony of Night :)

"TL;DR" Version: Check any "Top 100 Games of All Time" list, and shop at GOG.com.

First, keep in mind that I watched the "video-game revolution" happen. When I was a young kid, there was no such thing as video games. We had board games and card games of all kinds, but "game arcades" included pinball machines and their ilk. All physical and/or mechanical stuff. Then, one day, the "pencil and paper" games appeared and changed everything.

Dungeons & Dragons opened a veritable Pandora's Box of gaming, and many other PnP games followed, some of them still around, today. I would actually recommend you try a few of these as they are an experience unlike any other. GURPS, AD&D, and Paranoia would be a good start to getting your feet wet. There are also tons of amazing board games out, now. Many are old, but some are new. Try searching for "top ten boardgames" on YouTube to get started.

As for computer games, I've been playing those since they first hit the scene. The first ones were text-based and we played them on teletypes (something resembling a large printer with an attached keyboard). Once microchips had been invented, you started seeing things unheard of, before. A Pong console you could hook up to a TV. Video games in the arcades. The Atari 2600 VCS. A computer small enough to fit on a desk. Things just kept getting more and more amazing as technology advanced.

And now, after 40+ years of gaming, you ask me which games I enjoyed the most and won't forget.

Well, let me restrict myself to only computer games, and then only those titles which had the greatest impact on me as a gamer. For simplicity, I will try to keep them in chronological order. [Edit: Nevermind, that's too much work.]

"Collosal Cave/Adventure" -- The grandfather of all computer games, it spawned the "text adventure" genre. Definitely "old skool."
"COMBAT" Cartridge for the Atari 2600 VCS -- If you can get your hands on this, you have a party-in-a-box. Even now, this one cartridge brings more fun to your living room than the vast majority of today's titles.
"Mystery House" -- Sierra Online's first big success and the origin of the graphic computer adventure genre. Fun, but "old skool."
"Bard's Tale Series" -- The "Wizardry" series did it first, but Bard's Tale did it better by redefining the dungeon crawler genre.
"Ultima VII" -- The high point of the Ultima series. This series was a shot of adrenaline to the computer gaming industry.
"Commander Keen 4 & 5" -- Two of the best platformer games of all time, this series had a huge impact on the industry.
"DOOM" -- The game that put Id software on the map, the company that invented the FPS genre. The original is still lots of fun.
"StarCraft BW" -- Not the first RTS, but this game spawned "E-Sports." You might try StarCraft 2, instead.
"Diablo II LOD" -- A game I continue to play to this day, Diablo invented the "Action RPG."
"Planescape: Torment" -- Experience this game. Your life will never be the same.
"Baldur's Gate Series" -- This entire series is superb and single-handedly resurrected the CRPG genre.
"PLANETS" -- This amazingly addictive UNIX game spawned two genres: 4X and MMO. "GalCon Fusion" is a real-time game resembling it.
"System Shock 2" -- This legendary game was a "first" in many ways. You will never forget your first playthrough.
"Command & Conquer: Red Alert" -- Not the first RTS, nor even the first C&C game, but one of the best RTS experiences.
"Gabriel Knight Series" -- A legendary series among adventure games.
"Fallout 1 & 2" -- Following the tradition of "Wasteland," these two games set the bar for everything "post-holocaust."
"Master of Magic" -- Today's game developers are still trying to reproduce the phenomenal experience that this TBS offered.
"Star Control 2" -- Combining arcade action and a CRPG/space-exploration hybrid, this game was one of a kind.
"Monkey Island Series" -- The first three games redefined the point-and-click genre. The rest are good, too.
"Another World" -- An amazing game, you won't believe when it was made. You've got to play this.
"Neverwinter Nights" -- The original campaign was lackluster, but the modding community made this game legendary.
"Thief Gold" -- The origin of the "stealth FPS" genre, and still one of the best. The entire series is excellent.
"Sid Meier's Alpha Centauri" -- If you only play one Civ game, make it this one. Civ IV Complete is also excellent.
"Deus Ex" -- The first game of this series exploded our ideas of what an FPS could be. The latest game is also excellent.
"Star Wars Jedi Knight II: Jedi Outcast" -- This entire series was fantastic, but if you only play one, play this one.
"Beyond Good and Evil" -- An amazing and unforgettable action-adventure game.
"Psychonauts" -- A fantastic adventure/platformer game that slipped under many people's radar. You owe it to yourself to play this.
"Wing Commander IV: The Price of Freedom" -- This game and Privateer are still the standard by which all space sims are measured.
"Half-Life" -- Do I really need to explain this one?
"Unreal Tournament 2004" -- Perhaps not as fast-twitch as Quake 3, but more fun and laughs than almost any other FPS.
"Counter-Strike" -- Ground-breaking in so many ways, this game took E-Sports to a new level. CSGO is the latest iteration.
"Team Fortress 2" -- The most popular online game, period. 'Nuff said.
"Mass Effect 2" -- Arguably the best one of the entire series, this game pushed the boundaries of the CRPG genre.
"F.E.A.R." -- The first game of the series was so powerful an experience that the rest of the series couldn't match it.
"Bioshock" -- The first in the series and the one with the most impact.
"League of Legends" -- The only real competitor to DOTA, this game is easier to get into and free-to-play.
"Max Payne Series" -- If you like your FPS to shake your emotional core, this is your kind of series.
"The Walking Dead" -- Everyone else has already pointed out the powerful experience of playing this game.
"Plants vs. Zombies" -- Even non-PopCap fans got hooked on this one. A relaxing and extremely enjoyable game.
"Portal Series" -- Don't question, just buy it and play it.
"Rome: Total War" -- It's difficult to pick one game out of the Total War series, but these games changed the face of RTS.
"Minecraft" -- Addictive, endlessly fun, and highly mod-able, this game stands alone. The L4D2 mod is hysterical.
"X-COM: UFO Defense" -- Single-handedly responsible for a slew of remakes and spin-offs, the original TBS has yet to be matched.
"Chivalry: Medieval Warfare" -- Originally a mod, this now-for-retail game is FPS carnage. Prepare to get bloody!

There are a lot.
Bastion, Bioshock franchise (Infinite is just the best videogame experience i ever had), Far Cry, Final Fantasy VII, Darksiders Franchise, God of War franchise, Assassins Creed franchise, Deus EX:HR, Dead Space franchise, Dungeon Siege 1 and 2 (sorry but DS3 sucked so hard), Half Life franchise, both Portal games, Mass Effect franchise, Prince of Persia franchise, World of Warcraft and Diablo 2.

I can't remember more at this time, but hell, sure there are a ton more.

The games i missed in the comments (but maybe i have not red careful enough) :

Might and Magic 7

Boulder Dash

Final Fantasy IX i love that game i finish it on my ps1 like 4 years ago :D

Old Ones: Battlefield 1942, Mafia, Hidden and Dangerous 2, GTA III, NFS Underground, Half-Life 2, TES Oblivion, Age of Empire III
Some New: GTA IV, Battlefield 3, Dota 2, Crysis

Ace Combat 4.

Bonus game I'll never forget: Sid Meier's Alpha Centauri

Full Throttle was the first short game I'd ever played and still felt like I'd gotten my money's worth. The gameplay was sparse, particularly by today's standards, but the story and voice-acting was top-notch. Not sure where you'd get it these days, though.

The most recent experience I had like that was The Walking Dead series.

If I only could choose one game it would have to Mass Effect 3. It was the culmination of an entire franchise with all my decisions taking effect. All the friends I had met along the way and we teamed up for one big adventure. It was also the first game I ever bought on launch day. I nearly was going to buy it at the midnight launch but I waited until later in the day and walked into my local game shop and picked up my PC copy. I even asked the shop assistant about the "From Ashes DLC" but she didnt know what it was. Bit of a noob mistake. But yes, this game is currently my greatest ever gaming moment.

Bioshock 1
Mass effect 2
medal of honor: allied assault. (i cannot forget d-day)
so far, working on bioshock infinite, and it is damn good.

Dungeon Keeper

Elder Scroll series,
Dragon Age:Origin,
Mass Effect Series,
Fallout 3,
Fallout: New Vegas,
The Walking Dead,
Minecraft,
Saint Row 3,
GTA Vice City and GTA 4