Okay, so I've been thinking about this a lot lately after playing through a good deal of Sonic The Hedgehog (2006), namely because there are multiple flaws in it that prevent it from being as good as it could have been.
I'm not going to review it, I'm just going to address what I believe constitutes a good video game. These are mainly my own opinions, but I'm pretty sure that a few of them are common amongst people.
To start with:
GRAPHICS - When it comes to stuff like this, the graphics being realistic does not automatically mean that the graphics are good. Are they unique? Do they have their own charm to them? Are they visually appealing? This is one thing that I don't think many game companies understand.
Here's an example comparing two games that I happen to like.
HALF-LIFE 2: LOST COAST

MEGA MAN 9

Just from looking, it's easy to tell which of the two has more detailed graphics. Both of them are still visually appealing, but Half-Life 2 isn't quite as unique as Mega Man 9 is graphics-wise. It basically comes down to opinion, but either way, realistic graphics aren't automatically good:
SONIC THE HEDGEHOG (2006)

Okay, realistic graphics? Check. The level itself doesn't look very unique, but it looks okay. The real problem is Sonic here. His model is much taller than it should be. It doesn't sound like a problem, but in-game it looks horrendous. Sonic 2006 tried for realistic levels and got there, but save for the Last Episode it wasn't very unique in concept. It also didn't do so well with characters, which is more of an issue than it sounds from a visual standpoint.
A good example of realistic graphics that are still unique and visually appealing would be Super Mario Galaxy or Sonic Unleashed (PS3/360). Image time again:

Some of the concepts aren't very original, but in their execution, they look wonderful. In my opinion, this is the direction that video game graphics should go in. I don't mind FPS-style realistic graphics like in Half-Life 2, but I prefer the lighter, more cartoony style of Mario and Sonic games.
SOUND - This is both music and sound effects. Good sound effects can make the difference between a game feeling much more atmospheric and it feeling very false. I don't have much to cover here, just that the sound effect should fit what it was meant for, instead of sounding out of place.
Music, on the other hand - Well. Technology nowadays can run rock and roll like there's no tomorrow. Is this cool? I personally like Metal as a genre and have a lot on my iPod, but - Just because modern consoles can render it doesn't mean that every single game needs to use music like that. Example time again:
Mega Man 9: Tornado Man
That was composed on a music set that's somewhere between the NES and SNES. In the long run, it fits the level it's for and still sounds very appealing despite being based off of old technology. This doesn't mean that game designers should purposefully try to work under limitations like the sound chips of consoles from the '90s, but just because they can isn't a good reason to have rock, metal, or incredibly dark music for everything.
Once again, Sonic Unleashed example:
Chun-Nan: Dragon Road - Daytime
It's pretty light toned compared to music in other video games these days, but it still sounds nice.
PLOTLINE - Okay, I could really be going on for ages here. Nowadays, video games have incredibly complex plots that have no rhyme or reason to half of the events until much later on. This is okay in RPG's and a select few other scenarios in my opinion, but storylines don't need to have as much effort put into them as a lot of them do.
Example: The Half-Life series as a whole. Its plot is confusing and more tangled than spaghetti. Granted, it's a good way to keep someone playing, but... Seriously? Half-Life 2: Episode Two refers back to something from the first game that I don't think many people would've noticed unless they were paying attention, and Half-Life came out 10 years prior.
Sonic The Hedgehog (1991). It's got a basic story, and it's something that Sega did a lot back then, but it still explains the reasoning for why Sonic is doing what he's doing - Evil villain is trying to take over the planet. It's simple, and it's been done A LOT, but hey - if Mega Man 9 can get away with it, I think that other games can too.
Speaking of basics...
GAMEPLAY - THIS is what it all comes down to. I'm a retro gamer, so I'm going to state how I feel about this from that perspective.
Where to begin? The mark of a good game, gameplay-wise, is that you enjoy playing it, of course, but what defines what good gameplay is outside that? If you finish said game, do you feel good for accomplishing it or not? Well, to start out with:
*ENGINE - This is a bit more technical. This is basically the game's physics, its timing system if there is one, the player's control... That kind of stuff. To begin with, having glitches does not automatically mean that the engine is bad. Contrariwise, just because the engine has almost no bugs doesn't mean it's good. The question here is, "Is the game still fun?", not, "Is the game glitch-free?". I can think of at least one example.
Sonic Adventure DX.Yeah, I'm a huge Sonic fan The game's engine is incredibly similar to Sonic The Hedgehog (2006), but there's one big difference:
Sonic Adventure's gameplay is fast, has lots of action, and the physics feel almost perfect. The game is ridden with bugs, but it's still fun. In fact, I believe that Sonic Adventure DX actually has more bugs than its 2006 counterpart. The difference being that Sonic 2006's bugs were much bigger and more game-breaking.
*CONTROLS: ACTIONS - This one is a bit easier to explain, seeing as I usually talk with friends about how I feel on video game controls on a regular basis. In my opinion, the controls don't have to be any more complicated than, say, Super Mario World. You had four buttons - Two for run, one for a jump, one for a spin jump - the movement pad (d-pad), and the shoulder buttons to pan the camera around. There's nothing wrong with this, especially since the game is still easy to pick up and play. The game still has some twists to it - such as Yoshi - but either way, it's simple.
*CONTROLS: FUNCTIONALITY - Okay, this is a big one. I'm going to bet I'm not the only person who's ever played a video game and has felt like they're not in control of the character as they should be. An example right off the top of my head is most definitely Sonic The Hedgehog (2006). The controls aren't as complicated as they could be, but they're unresponsive a lot of the time. Normally, I don't mind this, but at several points the game throws you a twist and changes the feel to the controls, such as slowing the player down, not letting them jump, etc. I don't see anything wrong with this normally, but given the state of the controls from the start, it just doesn't work.
*LEVEL DESIGN - This is where simple controls work best. Nowadays, video games have to flat-out tell the player in-game how to play in case they didn't read the instruction manual. There's a serious flaw with this, as it shows that the programmers had little confidence in their design. Going back to the Golden Age of Gaming again: Super Mario Bros. (NES). The first level is essentially a tutorial without ever telling you outright how to play. Rather than explaining it myself, I'll just provide a link: "To the right, hold on tight". The person who wrote this knows what they're talking about.
OVERALL
The graphics do not have to be dark or realistic to be good.
The music does not have to be dark, metal, or anything like that just because it can.
The plot does not have to be incredibly complicated.
The control scheme does not need to be complicated to be fun.
The engine does not have to be glitch-free to be good.
The responsiveness of controls MUST be at least good.
The game does not have to tell the player how to play at any given time.
The level design should teach the player all the rules of the game.
The physics should feel right considering the setting.
The physics do not have to be ultra-realistic just because it's possible.
... Wow. I think that's the longest post I've ever made on a forum of any kind.

So... What do you guys think about this topic?