NES it would be when my mother used to babysit and all us kids would be playing Super Mario Brothers 3.

SNES it would be countless hours of Super Mario Kart battle mode with my brothers (miss the feather jump)

N64 it would be 4 player pretty much every game: Goldeneye, Perfect Dark, Rush 2049, Mario Kart 64, Super Smash Bros., Battletanx, Gauntlet Legends, Army Men: Sarge's Heroes...man those were the days

PC: Vette! w/ serial cable (chasing around all of San Francisco), Doom (had tons of WADs), Descent, Freespace, Command & Conquer (Red Alert was the best), Rainbow Six, the top down GTA and GTA2, Motocross Madness (tag was amazing), Half-Life (Sven Coop), Unreal Tournament ('99), tons of memories playing with my brothers on our local network.

I spend a lot of nights with my friends playing Super Smash Bros 64.

That's probably the most fun I ever had with my friends when it comes to video games!

The Nintendo 64 has the best memories
goldeneye
smash
Mario kart
duke nukem
pokemon stadium
Mario party the first one

Playing DragonBall Raging Blast 1 and 2 with my friend. Good times

Growing up my brother and I played a ton of Mario on NES and then SNES. For us doing the multiplayer thing, largely co-op, was just normal. I took a lot of it for granted in those days. Really only started to appreciate it when games like Secret of Mana allowed us to play co-op.

Secret of Mana was a big one. First RPG where co-op was more than just "you control these two in a fight I will do the rest" like it was in the SNES FF games. Mario Kart on SNES was a blast too.

N64 is where things of course got real good for us for multiplayer. 4 player Mario Kart 64 was a staple. I remember one game of Wrestlemania 2000 with my brother and two friends where one friend was so mad that he was losing he proclaimed "If I get hit with one more of those stupid fucking firedrivers I am leaving and not coming back!" Of course the three of us that weren't so pissed off promptly proceeded to just take turns hitting his character with the fire thunder driver move and he stormed out all pissed.  Other games like Killer Instinct Gold and Goldeneye were always great too.

PS1 we had a lot of fun with Dragonball GT.  My brother decided he would max out his crouching quick jab.  Sonofabitch took off full health bars with that move.

Xbox with the original Halo was a great time. 4 player split screen on a 20ish inch crt screen in my bedroom was awful, but yet oh so much magical fun anyway.

Later I would be big into Dark Age of Camelot, and even set up a few additional PCs in my bedroom so my friends could play in the same room when they were over. I still count it as we were in the same room despite it being a server-based game.

My brother joined the military shortly after high school and thus he hasn't lived local in the past 10 years. Whenever he comes home to visit though we have friends over and play Smash or SM3DW on Wii U. On his most recent visit we finally finished our "perfect" run through on SM3DW, hit the top of the pole with every character, got every start and every stamp, and never used the Golden Tanooki suits. That was a 3 year project and worth every hour put into it.

Playing all day in the Summer and coming in at night with my friend and playing Mario Kart on SNES. We got so good 150cc was no challenge and battle mode could go on a long time as we learned to avoid red shells. Remember doing ghost mode to shave off seconds to fractions of seconds on our best times handing the controller back and forth.

I'll give Nintedo credit for one thing in particular, they haven't forgotten that aspect of gaming (although I do wish MK8 had a proper battle mode).

Crash Kart racing, or whatever it was called. I was really young back then, and it's the only game I remember playing with my whole family.
Looking back at it, my fondest local multiplayer memories are from PS1 games

My first memories are of playing TMNT II and Contra on my cousin's Nintendo, which (along with all three Super Mario games) just blew my mind.

Otherwise, Road Rash II/NBA Jam/Toejam & Earl/Streets of Rage/Golden Axe on the genesis were a lot of fun, and of course there was perhaps the best local multiplayer system ever in the form of the N64. Mario Kart, Super Smash Bros, Mario Party...

and of course the memory of three friends of mine and I lying on our stomachs with our faces perhaps one foot from my old TV (the screen was about 12 inches by 12 inches) playing four person Goldeneye (so that screen that was almost the exact size of an Ipad2 was further split into four) is probably about the best memory I have haha, we somehow could see the game just fine and I feel I can safely blame that for at least some of the vision problems I'm beginning to have.

Otherwise, all the people that would come to our dorm room (sometimes upwards of 12 in a tiny room) to play games like Wii Sports and CoD 2/3/4 (3 actually allowed multiple local players online, was pretty cool) was just a ton of fun and something I like to think back on.

These days it seems almost dead outside of the Wii U. It seems like that focus may be holding Nintendo back, but I'm glad someone is keeping it alive.

It really was.  We'd have friends come over for big battles.  It was great.

Man, I remember the days of the serial cable.