I've been playing console games for a pretty long time, and am a very big console fan... all the way back to the Atari up through my 360.
Anyway, here are some games I constantly find myself caught up in even today.
Chrono Chross - PS1. This is an amazing RPG that warrants at least 3 playthroughs, and has 30+ characters to find within the game to integrate into your 3 man party.
Mortal Kombat (All the way up through Deception) Far back as SNES/ Genesis - Current consoles. This is one fighting game that has always been great to play with a group of buddies. Where else are you going to rip out your best friend's spine and not go to jail? Aside from it making Street Fighter look like Winnie the Pooh, it was the first game to utilize live-action actors animated through mo-cap in a game. The list of characters throughout the series have always been original and fun to play, although there is the occasional "What the fuck?". Stryker, anyone?
Resident Evil 1-4 - PS1, PS2, GC. I love this series. Zombies, mayhem, gore, things that make you jump - awesome. The most worthwhile part in the series for me personally is RE2. I love playing through two storylines, and romping through the ruins of Raccoon City is even more of a bonus. Lovin' it.
Splinter Cell - XBOX. Chaos Theory will eat your attention span for a week. Seriously. Playing as the uber special forces infiltrator extraordinaire, Sam Fisher, you go all over the world performing covert operations. It's a nice, stealthy, tactical approach to accomplishing level completions.
Kagero/Deception - PS1. Trapt (I know, lame name) on PS2. This is a demented game. That's why I love it. If you can still find the classic part 2, do so. One of the best classics you can pick up. The object of the game is this: Murder everything. You set intricate little traps, divided into three classes: Wall traps, floor traps, and ceiling traps. The key is to make them work in conjunction with eachother. It's twisted, but I'll be damned if it isn't entertaining.
007: Goldeneye: N64. If I need to say anything about this game in order for you to know why it's so enjoyable, step away from the controller. Seriously.
The Godfather: PS2, XBOX. Definitely a fun game. You go through old-time bank robberies, car chases, gang warfare... all the things that the classic Mob era what it was. Throw in voice acting by James Cannes, Marlon Brando, and Gary Oldman, and it's a well put together piece. You create your own character from the get-go, detailing him down to his shoes. The gameplay is just an enjoyment to behold.
Midnight Club 3: Dub Edition - PS2, XBOX. Don't let the name fool you. Sure, it's a stupid name if you're a racing fan. You know no one is racing a car on 20" rims. Way too heavy. Ignore that fact. This game has its share of obnoxious lame-ass riceburners, but you also get your hands on a ton of exotics and some damn fine classic 8 cylinder muscle cars. So just ignore the section of cars that sound like they're powered like a leafblower, and this is a fun arcade racer, with a very hands-on approach. You'll deck your car out from every facet of personalized body parts (Hood, bumpers, sidepipes, etc) to custom paint and graphics. And... yeah, you can put the lame-as-shit giant rims if you want.
This is already way too long, but I'll post a few quick ones that are worth your money: God of War (PS2), Prince of Persia: Two Thrones (PS2, XBOX), Oblivion (360), Hitman: Blood Money (PS2, XBOX, 360), Forza Motorsports (360), Call of Duty 2 (360).