Depends on the game.

They never did, really. Video games peaked, in my opinion, in the mid to late 1990s. Graphics have improved greatly since then, but the games haven't improved with them. I wish developers would spend less time, money, and energy on graphics and more time working on the craft of video games. Of course video games are a visual art, and the look and presentation of a game is important. But gameplay will always, always trump all.

Graphics for graphics sake; no. As a means of immersion and style; yes. Lately, I've grown fond of unconventional graphics and have found games like Borderlands 1 and 2 and Dishonored to be extremely appealing.
I think that a lot of gamers have a misconception that graphics need to be realistic to be pretty. Bastion and Muramasa are prime examples of this, both are gorgeous to look at.
I also enjoy the charm of the visual style of games like King's Bounty and Mount & Blade.
Then there are those technical masterpieces like The Witcher 2 and Metro 2033 which simply blow me away with their realistic and gritty style, there's room for every kind of art direction and looks in my book.

It doesn't matter, but It's a nice touch, it wouldn't matter more than a 0.5 grade if I was rating it.

Graphics are ok, but never a deal breaker for me.

I would rather have a less pretty game with sound framerate, cause there is nothing more aggravating to me than framerate issues.

I would love to see the same advancements for AI, that would be outstanding.

But its much harder to market AI or gameplay than it is to market graphics.

They matter to me more than they used to. Once I got my PS3 in 2011, that's when I started appreciating how great the games could look. The new NFS Most Wanted, for example, looks stunning. It's amazing.

Not graphics per se, but the art direction does bother me. Altough if the gameplay is really good, I will play the game no matter what.