Originally Posted by 4zb41

For me, I have to say first and foremost the game must be FUN. Following that, it must also be challenging. Sometimes the challenge is also part of what makes the game fun!

Fun, for me, can be had in a variety of ways depending on the game. Examples:

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1) I like Devil May Cry because it's fun stringing together combos and trying to look good while doing it - plus it's fairly difficult, with the variety of different enemies of varying difficulty escalating as the game progresses. They demand different approaches to different battles, meaning you have to change your technique to suit the situation.

2) I have fun playing just about any co-op game with friends. Mostly because of the banter you exchange with your partner(s) while playing it. Army of Two was great for this as you could brofist your partner for a job well done or smack him over the head for being an idiot - usually followed by a lot of friendly yelling at each other and being chased while your partner tries to smack you over the head.
Rainbow Six: Vegas 2 also lead to several hilarious situations in which my partner would often refuse to work together and ran off and got himself killed several times. In the end we ended up using him as bait to lure the enemies into C4 traps.
Lost Planet 2 also ended up being crazy fun in co-op because of the fact you and your partners could all fill different roles in a fight, you could fight over any good equipment that was dropped - or mechs - and generally smack each other around for being stupid. There was also no "KEEP CLOSE TO THE OTHER PLAYERS OR ELSE" feature, which was good.

3) Sheer hilarity - whether that be the result of a wonky physics engine or the game's style - is often a selling point for me. If playing the game has me laughing regularly, I consider that to be fun. Good examples would be Gmod with its abuse of the source engine, and No More Heroes in which absurdity is an every day occurrence. I'm going to go ahead and mention the Space Channel 5 series here too, partly because its English dubbing is "so bad it's good" and partly because of the absurdity of everything involved. "Let's beat the bad guys with our superior dancing skills! ...and lasers. Maybe." Note "bad guys" ranges from other humans, to dancing robots, to gigantic mutant plants, to humongous mecha.

4) "Awesome Factor" or "Holy Shit Quotient" is also a pretty good measure of if I'm finding a game fun. The HSQ (thanks tvtropes) is measure by how many times you say "holy shit!" whilst playing the game - in a good way, not a "dude wtf this game's bullshit" kind of way. Good examples of this for me were Shadow of the Colossus (at least one per battle,) and Armored Core: For Answer in which every battle against an arms fort involved dodging humongous projectiles bigger than yourself, and systematically tearing apart what is essentially a gigantic moving city bristling with cannons, HMGs and missiles apart piece by piece. I suppose I just like fighting against gigantic things in games.

5) The challenge itself is somewhat makes the game fun. Dark Souls for example is certainly not easy, as you will die many, MANY times during the course of the game - but it's fun because as you level up and become more skilled, as well as mastering the controls and your favourite weapon's moveset, you begin to actually triumph over everything that caused you difficulties before, which feels immensely rewarding. In the same way, (almost) every win in a boss fight feels like an accomplishment as they no doubt killed you several times before you finally managed to best them.

6) And finally, character customization makes a game fun for me. I'm not talking so much what they look like (you probably won't see their face anyway) but rather how you can level and build your character to suit one (or more!) of a variety of roles. Mount & Blade is one of my favourite games in this regard. You can play however you like, depending on how you spend your levelup points. Make you character based on trading, speechcraft, and leading troops in a world where you almost always have to throw yourself into combat? Sure. Want to play the Nordic berserker who runs into combat wearing nothing but a loincloth and swinging an enormous battleaxe? Yep. Feel like barking orders to your troops from on high whilst taking pot shots at the enemy from afar with a crossbow possibly hitting your own guys in the process? You betcha.
Playing the hero who charges in with a lance on his shining white stallion and single-handedly wipes a small battalion off the map is also entirely possible.

TLR:
Devil May Cry because it demands you be "stylish" with your combos and mixes up the enemies per battle to demand a different approach each time. Also because it's really quite challenging, and has a good sense of humour.
Co-op games more because it leads to hilarious situations in game and generally exchange of friendly banter between you and your partners if you're playing locally with friends. Examples were Army of Two, Rainbow Six Vegas 2 and Lost Planet 2. Also Splinter Cell: Conviction's co-op side story, now that I think of it.
Hilarity or Absurdity in a game is part of what may make a game fun for me. Examples were Gmod and No More Heroes. Also anything with a fun physics and/or destruction engine a la Red Faction Guerilla or Battlefield: Bad Company 2.
Holy Shit Quotient or how many times you say "holy shit" over the course of the game. For me, this usually means the game is fun because it involves fighting - and beating - humongous bosses with your tiny character. Examples were Shadow of the Colossus and Armored Core: For Answer
Overcoming the game's difficulty makes a game fun for me too. Example was Dark Souls - the things that curb stomp you at the beginning can all be flattened by you later after you become more skilled at the game and level your character.
And finally, Character Customization. Being able to customize my character to fit any role I choose is a good selling point for me, especially if there's absurd options available. Mount & Blade was this one's example.

The more features mentioned above that a game contains, the better IMO. A game that combines hilarity/absurdity, local co-op, a fun physics engine, HSQ, extensive character customization and at least a decent challenge would be a welcome addition to my collection. I'm thinking a combination between Dark Souls and Red Faction Guerilla with drop-in co-op and a writer with a sense of humour.

Originally Posted by JellyNinja

Which brings me to another point I would like to drag up here, "feeling" with the character you are playing with, running alone in a grim world filled with monsters, and the only living humans there do not give a shit about you.
I may be wrong here, but yeah, felt lonely in that game.

Originally Posted by BlubKill

multiplayer games are more awesome if u play against people which should be better then u.

Originally Posted by 4zb41

This is kind of an odd point for me, myself. If I go and put myself on online multiplayer and get absolutely roflstomped constantly by everyone, I'll get frustrated and think the game isn't fun at all.
HOWEVER it does become more fun if I persist with determination and eventually work my way up through all the difficulty so that in the end I, myself am the one doing the roflstomping.
Not to mention how hilariously fun/rewarding it is to actually kill someone who has been stomping your ass the whole round. Bonus points if you did it in an underhanded way.