Ah, the good ol' Nintendo 64. There were lots of great games for that console. Here's some of my favorites:
Super Smash Bros - awesome fighting game, particularly if you invite some friends over. I actually prefer this version over Melee and Brawl due to its increased knockback damage and downright bloodthirsty crowd (seriously, they're WAY more excited than in the sequels).
Space Station: Silicon Valley - a quirky little platformer where you take control of various robotic animals, each with their own attacks and abilities. Developed by the same folks that would later make the 3D Grand Theft Auto titles.
Paper Mario - originally meant to be a sequel to Super Mario RPG, Nintendo eventually decided to go a slightly different route and ended up making this gem. While more of an adventure/platformer with RPG elements, the battle system is very well done and keeps the player engaged through "action commands" that require various types of timed controller inputs to improve the performance of attacks and minimize damage taken. The game also has a great sense of humor with a few moments that elicited genuine laughter from me.
Mario Kart 64 - still my favorite in the series. Play a few Vs. and Battle rounds with your friends and bask in the chaos that unfolds.
Banjo-Kazooie - Perhaps the pinnacle of Rare's 3D adventure/platformers. It has expansive environments, solid gameplay and a great sense of humor. It also has just enough item collection without becoming tedious like Rare's subsequent games (*cough*donkeykong64*cough*). Just be warned that it's a difficult game, particularly towards the end.
Star Wars: Rogue Squadron - while not as deep as the X-Wing series for the PC, Rogue Squadron still offers lots of entertainment via a more arcade-oriented approach to starfighter combat. There's a wide variety of missions, and all of them have plenty of Imperial targets for you to shoot. If you manage to beat missions quickly and efficiently enough you can unlock some bonus vehicles and missions. There's also plenty of awesome cheat codes that unlock even more, including a flying Buick that shoots cluster missiles.
Star Wars: Battle for Naboo - made by the same folks that developed Rogue Squadron (Factor 5), this is a similar game, but set during Episode I. New innovations include being able to switch vehicles during the middle of a mission via special hangars and ground-based vehicle combat. Regardless of how you feel about the prequels, Battle for Naboo is a fun game (besides, the intro screen has the N64 logo crushing Jar Jar Binks. That alone should be worth the price of admission for most Star Wars fans).
Star Fox 64 (or Lylat Wars in Europe) - a flight-based rail shooter, this is essentially Nintendo's answer to Star Wars. As mercenary flyer Fox McCloud, you must save the Lylat system from the evil Andross. While you can breeze through the game in about an hour or so if you're experienced, there is extra replay value in the form of branching paths with different levels to tackle (along with being able to see the "true" ending if you go the hardest route). Just remember to "do a barrel roll!"
Bomberman 64 - Okay, I'll admit that one of the main reasons why I included it is the soundtrack, particularly the score for the Blue Resort level. Still, the game itself is fun, particularly multiplayer. It can get pretty insane with four players all trying to toss and kick bombs towards each other.