It seems like a simple question right  on the surface, but there is quite a lot of depth to this question. What aspects of a game are actually what you motivates you to play it. A while back I knew that Final Fantasy Tactics was my favorite game, but I didn't know why. I just assumed that it was because of the game play. So when I went to look for other games to play I looked for other tactical or strategy games like Disgaea, Fire Emblem and the like. None of these games were the game I was looking for. They weren't bad and I enjoyed playing them, but they didn't make me care about the characters or feel invested in the world. So I sat down and looked at what about FFT is what actually made me like it so much. It was how they did characterization in the game.

In FFT the characters all feel like their own people. The first villain I ever hated is in this game. When an NPC with 6 lines in the game is killed I cared because they mattered to the other characters. A little over half way through the first chapter in FFT there is a scene where the hero and his best friend are sitting in a field because their problems are too big for them to handle. Most any other game the hero is fearless and isn't slowed down by anything, but here you are just a bunch of kids with problems far bigger than you are.

So now when I want to play a game like FFT I look for story games with really strong characters instead of tactical games and I find myself enjoying them more.

So what about games makes you like them? Is it the gameplay, the story, the characters, the aesthetics, the challenge of competing with others or beating your scores, or is it just the fun aspect?

This will probably vary for you on a game by game basis because everyone plays games for different reasons. When I play monster hunter I am doing it for the gameplay. It is really fun to fight giant monsters from time to time and learning how to fight a new monster feels really rewarding.

The Last of Us has an engaging story that really makes you want to see it through to the end.

Portal is a really fun game because of the characters. With no GLaDOS the game would still be fun but it wouldn't be the game that it is. It would just be a puzzle platformer with a unique mechanic.

Journey is a game that is worth play just because of how amazing the aesthetics are and how fantastic it makes the world.

When I play Smash bros or League of Legends I'm doing it because I like the challenge of facing off against other players or trying to beat my time trial by another 3 seconds.

Please let me know what games are your favorites and why you like them more than other games in their genera.

I like the games that I do based on lots of factors. If the story's engaging I would be more inclined to beat the game, if the game play is fun I would play it more, and some nitpicks things as well. Like the character design, if I don't like it, I wont play the game as much as I would any other game with a design I like. The challenge has to be in the middle for me to not get bored with it or rage quit on it. There's a lot more, but that's the main points that I look for.

tiropat : I like the games that I play because most of them have great storylines, great gameplay, great soundtracks, are extremely addicting, good multiplayer, or in some cases all the above.

I tend to the like the games that I do for various reasons. 

I tend to like Halo because it was something that I got started in with my friends. I like because of the story. The story is one that I can really wrap my mind around. 
I tend to like Call of Duty games mostly because of Zombies and Survival mode, but I also like how easy the multiplayer is. I mean, I may not be good at it, but it's not something that you have dedicate time to. You can just go online, play a few matches, and be done. It's not like you have to play 50 games and then you are satisfied. No. You can play 3 matches and be happy. 
I tend to like racing games because growing up, I always played them. I always liked going fast. I don't like to be stuck behind other people. I always enjoyed go karts growing up, so that could be another reason why I really like racing games. 
And there are a few R P G's that I like.  I guess I like those because I feel a little free when I am doing them. I am not rushed to do anything, and I have more freedom than i do in other games. 

tiropat :  I like most games for their exploratory features.  However, I wanted to comment on your feelings about portal. I actually love the game for its puzzles, but I also completely agree with you. The story is so attractive and well done that it's hard to imagine liking the game as much if the plot was missing.  Although I'd still enjoy the game without that characteristic, there's just something absolutely wonderful about the whole package.

Singelli : That is exactly the point I was hoping people would understand. Every game gives you one reason for liking it and wanting to play, but the truly good games have layers upon layers for enjoying them. Every Final Fantasy game has it fans but the ones that are truly great have fun combat, characters you care about, and a world that both encourages and rewards you for exploring it. If you take away any aspect of the game then it goes from a great game to a merely passable one.

I've really gotten into Skyrim and currently playing Dragon Age: Inquisition I like games
like these because it puts into perspective of Medieval times and I'm weird and pretend that
the guy im playing as is me lol but i like games like those because you can explore kill dragons
doing certain missions mold the way your guy becomes its just fun and adventurous.

Thats about all i can say i don't have any other reasons that i can think of at the moment.

tiropat,
I won't play FF games because I avoid all games with magic. However, I agree with you about the best games having 'layers' of likability. lol  There are TONS of games out there with jaw-dropping graphics, but I've also played great games with cruddy graphics.  It must be hard to find a decent balance of visual appeal, great plot, player-friendly mechanics, and attractive goals / missions.

Minecraft and Spore are probably two of my favorite sandbox-ish games because... while I get to explore to my heart's content, there is still an element of danger which is not overwhelming to me.  The in-game risks put me on edge to make it exciting, but don't make me rage-quit in frustration.  The perfect balance and nearly-limitless gaming ability are absolutely suited to my gaming style.

Singelli : I think Minecraft is a unique game in that it is actually several games all wrapped up into one package. If you are playing on a multiplayer creative server you are not playing the same game as someone who is playing on a hardcore survival map. Solo survival is way different than multiplayer survival because in solo you will always be able to find some animals for food, but online survival they are all killed or hidden by other players. Even the amount of protection you are able to give your stuff online can result in very different gameplay, if you can claim land and lock chests you will be less paranoid about where you keep things than on servers where you can't.

Minecraft is really great in that if can bring you in for creativity, exploration, and survival and give you different amounts of each depending on how you want to play the game.

Why do I like the games I like, well, these games do quite interest me a whole lot and 7/8 of these games I do happen to play I have grew up with and I'm never too old to give them up. Like the Mario series, Pokemon series, and just to name a few series I grew up with that I still play today.

They immerse you in another world that you are directly involved with, and  you have a direct impact on that world. Not to mention that video games are simply fun.