Atari 2600: 

Mario Bros.:  This game is NOT the popular old Nintendo game you've probably played.  Instead of a side scrolling game, it is a game full of platforms where you must kill all the enemies to beat a level.

NES: 

Super Mario Bros.:  Simply a classic.  This 1985 work of art is the best-selling game of all time, and in my opinion, the best game of all time.  It was the first majorly popular sidescrolling running game, paving the way for many other great classics.  I won't say anything else about this one, since most people have probably played it at least once.

Castlevania:  Great game about a hero named Simon Belmont who kills monsters in castles.  It is a pretty difficult game toward the end, and the game's final boss, Dracula, is considered by some to be the hardest last boss in video game history.  Did I mention I can beat him?

Contra :  A great 2-D shooter and a fun challenge all the way through.  If you don't use the famous cheat at the beginning (Up, Up, Down, Down, Left, Right, Left, Right, B, A), this game is incredibly hard.  If you do use the cheat, it is still fun and challenging.

Super Mario Bros. 3:  Probably the hardest Super Mario Bros. game, another great classic.

R.C. Pro-Am:  The best racing game of its time, one of the best racing games ever.  Incredibly fun.

Jaws:  Driving a boat, swimming, and shooting at a shark.  A very unique and fun game.

Sega Genesis:

Sonic the Hedgehog 2:  Pretty easy, but very fun.  If you're into challenges, try getting all the chaos emeralds and getting Super Sonic.

Gunstar Heroes:  The best co-op game EVER.  Defend the galaxy in this awesome 2-D shooter.

Super Nintendo:

Super Mario World:   Yes, I like Mario.  Another great one, this game is larger and more complete than the previous Marios.

Donkey Kong Country:  A brilliant 2-D sidescroller.  Just strategically brilliant.

Killer Instinct:  The best fighting game EVER.

Mortal Kombat:  The second best fighting game ever.

Zombies Ate My Neighbors:  The second best co-op game ever, this one is a 3-D overview where you walk around, rescue people, and shoot zombies.

Nintendo 64:

Super Mario 64:  This game is great.  I strongly recommend it.

Goldeneye 007:  The first great first person shooter.  This game was groundbreaking.

Star Fox 64:  Fly around in an arwing, shoot enemies, and defend the Lylat System.

The Legend of Zelda:  Ocarina of Time:  In my opinion, the best RPG ever.