Doesnt really matter to me, but I do admit that there comes a point that it can dampen the experience. For example, I love ff7, but it is incredibly hard for me to play these days due to its blocky graphics. It has not aged well at all. In fact, I think the snes sprite based games still look way better...

I selected "The are a big factor...", but perhaps I should have selected "Other", because what is important to me is atmosphere, and the visuals of a game are most of that. So graphics, yes, but not graphics performance. For example, these visuals are still some of the most athmospheric for me even if the graphics is basic to say the least:
.

Great graphics are never a necessity for me, but it's something i always appreciate. Having a beautifully rendered world coupled with an appealing art direction can add a lot to my enjoyment of a game. Good animation in particular is something i put a lot of value in.

I tend to value the IQ side of things more than in-game visual fidelity though. I'd rather the latter be worse if it means i can have a pixel perfect resolution and a nice AA method attached (which is what i always target on my PC). Great graphics don't mean too much to me when my attention keeps being pulled away by blur and jaggies :p

As an quick example, Elder Scrolls V: Skyrim. Some parts of the game looked good and other parts after looking at it closer look horrible. 

In the opening sequence cave, the cobwebs had pixellated edges and were on a completely flat plane. It looked like the kind of thing you'd find in Duke 3D.... a pixellated  sprite with a transparant background that your character clips through as if they weren't even there.

In the golden claw cave, on thee other hand? The cobwebs looked just fine. 

I'm sure you played plenty of games that were like that and didn't notice or cared.

Thanks for sharing that. : ) Unfortunately I haven't played Skyrim, so I am not familiar with that part, but I imagine that little detail may have been glaringly noticable once you see it. I can see where you're coming from. It's possible I have seen this before in games I played, but I may not remember them, or it didn't bother me too much..

I never played FF7 when it first came out, so when I tried to play it, I had the same issue... : / I think I couldn't get past the graphics last time I tried it. Maybe if I give it another shot, I can enjoy it.

How is it cheating if he draws exactly what he described to me?

I prefer HD with stable frame rate.

Graphics just like Audio, pacing, directing, level design, controls, etc are just one piece of the puzzle of what makes up a game. Without graphics there is no game - period. But sometimes they aren't as important as they are in other games. The Last of Us required beautiful realistic graphics to convey the story and characters, Minecraft has no need for these things.

Also "Good" graphics and "Realistic" graphics are not the same thing. For instance Fat Princess had great graphics, and so did Killzone 2 - just in different ways.

I think art style is also a big factor for me since many of the games on the PS4/XB1, while being graphically impressive, just don't pull me in so much. They're just not so interesting to me. It's likely I am not so crazy for realism. Games like 3D World and Smash did get me really excited to play them, however.