Gaming with roommates

My go-to group local games are: Sonic All-Stars Racing Transformed, Broforce, Castle Crashers, and any fighting games (Nidhogg, Injustice, Mortal Kombat, etc.)

Worms Armageddon
Pocket Tanks - It's 1v1, but you could just all take turns

Cards Against Humanity - This is a physical card game, but it's great and it may be something that you might want to check out

I can't think of anything for six people at once, but i second the Broforce recommendation! Me and my friends had a ton of fun with it. It even has a free The Expendables spin-off. CLICK!

Overcooked (4 player max)
Rocket League (4 player max on one screen, + more in LAN setup)
Starwhal (4 player max)

Also, one of my best memories of dorm life was running LAN parties and tournaments of things like Quake 3 Arena or other such similar games that can be set up for deathmatch/capture the flag/etc. Since you are all already on the same network, it's really easy to set things up for LAN, and you can all game in the same space too pretty easily if some of you are on laptops.

6 players will be hard to find, only fifa comes to my mind and I don't like that game at all.

I'd suggest Overcooked which was already recommended above, Worms is a classic and you should be able to fill it even with 6 teams at the same time (or just do 2v2v2). I'd recommend more but I didn't play many more than these, you'd definitely have more choices if you instead wanted just 6 players on 6 PCs and not a single screen.

Gunscape - 8 players split-screen
Raining Blobs - 8 players split-screen

then you have fighting games where they can take turns to play.

Mario Kart is always safe bonding, also an excellent excuse to murder a roommate you don't like
But any of the jackbox games will work honestly, although murder party is the best game from it in my opinion

Towerfall Ascentios or Invisigun Heroes are great, but only up to 4 people - but really fun to watch, and one can set up short rounds to swap. I don't know what supports six, but I wholeheartedly recommend these ( think Duck Game also fits the 4-player base)

Keep Talking and Nobody Explodes

For board games, Secret Hitler is best played with 8 or more people, but fights may break out.
There is also Resistance, which is made by the same people that made Secret Hitler. It's playable with 5-10 people and has a similar style.

I don't know many one-screen multiplayer games, but I can definitely suggest you the Trine trilogy and Broforce. Unfortunately, they only allow up to 3 and 4 players respectively.
There are many turn-based games like the Worms series (there is a free, open source Worms-like game called Hedgewars if you want to try without spending money) or turn-based strategy games like Battle for Wesnoth (excellent free, open source strategy game, which I recommend mostly for single player campaigns). The problems with these games, however, is that they are a bit slow (especially Wesnoth), so I wouldn't recommend it for more than 2 players.
Another game (that I haven't unfortunately tried myself yet) that seems very good for 2 players would be Divinity: Original Sin.
Otherwise, I'm sure there are some sports/race games that support 6 players, but I don't really know, since I don't like that kind of games.

Wow, some great recommendations here. I've played Secret Hitler before--it was fun and they have a free print to play. Although already mentioned, Mario Kart is the first that comes to mind and then there is rock band, super smash bros and many others like it. Although it doesn't allow for more than 2 players I would also recommend Portal 2.

Edit: Just came across this: Deceit is going free to play and it's for six players (but I assume each player would need their own screen).

My roommates and I all play Minecraft (modded, of course). We played Terraria for a bit as well.
Any MMO is a good pick too, ours was TERA. Before that, it was WoW.

keep talking and nobody explodes is really cool for couch coop.

I'm very big on local coop, it's my favorite:)

Unfortunately, more often than not, these type of games are lackluster and bland. Some are great, though.

I've played most of the Jackbox type games, including Use your Words - I recommend Jackbox Party Pack 2, it's the best of the bunch. I 100% recommend it. Perfect for 6 people.

Overcooked is really fun and challenging. (4 people)

Games like Towerfall Ascension and Duck Game are also really fun, short games, where you can switch players fast, winners stay or whatever. (4 people).

Eon Altar is often overlooked, but it's a great game. An RPG where everyone has a character, and controls stats, inventories, etc, from their phone. Then you must work together in turn based tactic style battles. I guess it's a little more serious. (4 people)

Personally, for me, the best local coop experiences I've had were story driven games (Beyond 2 Souls, Telltale, Life is Strange, soon-to-be Detroit: Become Human), where we all huddle on couches and watch the TV and story while one person plays. I feel these games were meant to be enjoyed with people.

no one mentions Gang Beast and Speedrunners yet?
ohh ohh Brawlhalla is fun too, and its free

Steam:

• The Borderland series
• GrimDawn
• Divinity: Original Sin
• Divinity: Original Sin 2

Non-Steam

• Diablo 3

Talisman and all its many DLC are always good for couch co-op. It is up to 6 players, though if you did local multiplayer hotseat, everyone will be able to see your cards. Still a fun way to play, though.

Tiny Brains is on sale here (max 4 players)

Dont starve together is a good coop game to try.

Screencheat is fun, but limited to 4 players if I recall correctly.

Not steam, but great curvefever! It is always funny to play it with friends on one computer. :D (and it is possible to play up to 6 players).

I'd suggest browsing the local-coop tag on Steam and using the search on SG to search for past threads, you might find some stuff you like.

I hear Invisigun heroes is good.
The Jackbox games are pretty good if you are not playing with people with English too poor to keep up with the game like my Mom's. Then it's just frustrating for her and, in consequence, for everyone else.
Jackbox 1 and 2 have been available in some bundles and there are some sales where you can get 1-3 together for like $30.