For me...
All games have something they're TRYING to do. Some sense of enjoyment that they're aiming for. Some focus.
If they can't do well at what they're focusing on even at the most basic level, it's awful, to the point of being unplayable. Examples of this include Heavy Rain and Final Fantasy XIII, which fail at basic storytelling, story being their area of focus; Dragon Age 2, which identifies itself as a "strategic RPG" but has less strategy to it than an average Gears of War combat scenario due to the "wave"-based encounter structure, among other things; or Heavenly Sword, which was clearly built without any basic knowledge of how to develop a good hack-and-slash and lacks any solid sense of challenge or structure apart from a couple of vaguely interesting ideas. They each have something that they're purported to do, they just don't do it well, and there's clearly better places to go to fill the niche each of these games are meant for.
If they meet that focus, do it well, and build on it, then it's good. Super Meat Boy is a good game, picking a simple focus--pure platforming-based puzzle-solving devised as a throwback to oldschool platformers, only evil--and following through with it, developing 6 worlds of diabolical challenges, plus "Dark World" versions of those worlds for players looking for a challenge, plus "warp zone" levels with hidden characters and other secrets hidden in each stage, plus an entire bonus world. Everything they could have done with Super Meat Boy, they did. I'd be hard-pressed to find a better example than that.
Average? How to describe an average game. I see so few of them, gaming has become very black and white to me. I'd say that an average game is one that identifies its focus clearly but has rough parts or doesn't build on its focus. Ninja Gaiden 2/Sigma 2 is a good example. It's got solid fighting mechanics and risk-reward elements, but the gameplay is extremely one-note. I get this from Doom 3, Crysis 2, Uncharted 1/2, Call of Duty, Borderlands, Fallout 3, God of War 1/2, Halo, and the Mass Effect series as well. They all have a very uniform feel to them, lacking in variety or surprises. I feel like if I've played thirty minutes of them I've seen just about everything they have to offer. The short campaigns of modern military shooters are a testament to that lack of room or foresight for growth; the simple truth is that they're so plain and so limited by their level structure that developers can't generate more than four or five hours of content--even if they have the time and funding to do so--without it getting painfully redundant, like it did with Doom 3. Hell, it didn't take six hours for me to start to hate playing Uncharted 2.
Below average... take what I said about an average game, and fuck up some basic element that a game shouldn't fuck up. Devil May Cry 4 is below average. Yes, it has its focus--the same style-based combat as Devil May Cry 3--but it fucks up basic game flow by forcing the player to go through every level and every boss a second time backwards with a different character instead of having a proper progression through new levels, bosses, and content. Not very satisfying, rather lazy, and definitely no substitute for proper content. The Force Unleashed has the same problem, and The Force Unleashed 2 to an even greater extent, having more than enough potential to develop a full plate of content but somehow just not serving it up.
A GREAT game surprises me by pushing the boundaries in some way. Boy are those hard to find. You see people trying at this a lot but really just failing at basic shit and making excuses for why they fail at basic shit, like a lot of the games I defined as "bad" above. You see a lot of people putting together average or below-average games but putting a nice thick coat of spectacle on them and trying to pass them off as great, but more often the greatness of a game is under-stated. Batman: Arkham Asylum is a great game, not only delivering a near-perfect realization of Batman in game mechanics but also employing unique four-dimensional level design, with segments of Arkham Island changing frequently and presenting new challenges as Batman gets to the bottom of the Joker's plot, bringing the world of Batman to life where it could have settled for just Batman himself. Dungeon Keeper is a great game, providing a full range of rich and engaging experiences that emerge purely from the act of playing the game. In a playthrough of one level you'll mine for gold to build your defenses, discover a hidden underground river leading to the entrance of an abandoned temple, and send an expedition of minions to navigate its traps and recover its treasure, only to discover hidden evil that's been sealed away for centuries... which you now control. And that's just one emergent story in one level.
Things that set me off personally...
Lack of flexibility, above all, sets me off. If I feel like I don't have room to play the game, I get frustrated. Uncharted epitomizes this peeve of mine, presenting me with a walking tour of its levels with occasional shooting gallery segments instead of giving me the sense of actually being an explorer or a treasure hunter. The whole experience feels like I'm jumping through the developers' hoops rather than engaging in any kind of creative problem-solving, to the point that the game has decided for me exactly which weapons it wants me to use and exactly what order it wants me to shoot enemies in.
By contrast, Metal Gear Solid 3 gives me locations laid out as they would be if they were real places, gives me a goal, and says "I don't care how you get there, just get it done. Bonus points if you don't kill anyone or get noticed." Crysis 1 gives me a lot of this as well. Devil May Cry 3 has a lot of things that should set me off, but there's enough detail and content in the fighting engine alone that I've got plenty of room to get creative with how I dispatch enemies, like the Tony Hawk of hack-and-slash.
Bad storytelling gets to me too. Bad writing I can take. Clunky dialogue or shallow character motivations I can take as long as I can understand what's going on and the story gets moving. Cliches I can take as long as they're plausible and don't come off as pandering. But bad storytelling? Just plain being bad at communicating what's going on, why I should care, or what the story is even about? That sets me off. Heavy Rain is one of those games, unable to find enough focus between its four characters to bring any of their struggles full-circle, often coming out of nowhere with essential facts and depending on cheats to artificially maintain suspense. Bioshock is another one of those games, building on one set of themes only to discard them for another before a resolution can be found, taking its twists and its morality system too far when all it really needed to do was pose the player a question: How much of a monster are you willing to become in order to escape this place? As it is, it feels like it's trying way too hard to impress me.
By contrast, let's look at Final Fantasy 7. Sephiroth is--let's be honest here--a poor villain taken on his own merits. His megalomaniacal turn comes from weak foundations and his oedipal complex is purely ridiculous. On the other hand, though, he has a good image, is an imposing antagonist, and serves his purpose as a representation of the Shinra corporation's reach exceeding their grasp, prematurely meddling with powers they don't fully understand as a natural progression of their meddling with everything else. Point is that he ain't realistic, but he does what he needs to in order to make the story work, make it interesting, and push its themes full-circle. Another example is Crysis 1. Straightforward story about aliens. It's not anything special as a story, but then again how special did it honestly have to be? They build the aliens up effectively as imposing antagonists, the Korean military cell occupying the island make an effective contrast and establish a sense of context compared with these aliens, and both make for a variety of interesting problems for the player to solve. It's an under-stated story that does what it needs to do to make the game interesting, and that's all I ask for.
Welp, as I've probably written several pages by now I think I'll just stop here. To sum it up: whatever it is the game does, I'd just like it to be good at it! Not sure what you wanted to get out of this, but I hope it was useful to someone.