Graphics are a brilliant addition to solid gameplay.

Pretty much all Nintendo games (except on wii-excl Mario Galaxy) were sold to me on the appeal of their graphics and me wanting to jump into their world.

Graphics are important to me in a manner that they need to be realistic, I simply cannot play a game with a cartoony look. The most cartoonish I can tolerate is Saints Row 3/4 style of graphics.

But as long as I get realistic graphics, I really do not care if they are 720/900/1080p, any is fine, I've played 5 years on X360 games that are perhaps even sub-720 (and I really didn't know or care what exactly they are). A great gameplay in 720p always beats a great-looking (1080p) game with poor gameplay (The Order, for example).

I will ask my DM to draw a picture :P

after 15 minuties playing, they dont matter.

but they are needed for selling the game.

I do mind the graphics for a game, but for me, they are the second most important factor after the gameplay. It's not like I prefer to sacrifice gameplay for ultra realistic 120 FPS 4K and maxed AA, but I don't want jaggies, muddy textures and distance pop up. It kinda ruins it for me.

42.67% I love good graphics, yet atm I'm addicted to Minecraft again.
Art style and 'running well' are more important than the latest technical bells and whistles.

Depends on the genre really, For rpg's i couldn't care less of the graphics i'm happy to buy inde or rpgmaker games on steam as long as they are good, but if I were to play a FPS I want the graphics to be as amazing as possible to get that "WOW" feeling.

Wether the game is fun is what is the most important for me.

With that said, its not like graphics are completely overlooked. Something that looks 8 bit like alot of today's indies, i wont even consider. There are limits to what my eyes will tolerate and graphics do factor in how enjoyable the experience is, but they certainly arent the main requirement. Just dont make it revolting.

Yoshi's Wolly World looks gorgeous and "realistic" if you can strech your imagination a little bit.

Graphics are not the end all be all for me.  I play video games to have fun.  I don't need to forget I'm playing a game and feel like I'm watching a movie.  If that were the case, I'd watch a movie instead.  And even then, the same applies for me.  My brother told me that people "got lost in the world of Avatar and felt depressed leaving the theater as they re-entered reality."  My experience watching Avatar was very different and far more disappointing.  Just because the movie had top of the line CGI didn't change the fact that it was still Dances with Wolves set in space.  I'd also prefer to watch the original cartoons than the CGI remakes of Scooby-Doo, the Smurfs, Alvin and the Chipmunks, etc.  Anyway, back on the topic of video games, if I'm plunking $60 down for a game, it better be fun and last longer than 6 hours.  Because I can buy three 3 hour epic movies for that price and get far more enjoyment (probably more replay/re-watch value too) than a game that relies on visuals alone as its selling point.  Don't get me wrong though.  If the game has the whole package, then that's great, and I'm all for it.  But, graphics should take a backseat to gameplay (the whole purpose of "playing" a video game), and story.  It should enhance those aspects, rather than trump them.