what is the best chess game ever played in history?

Bobby Fischer.

If you mean best by both sides, then it would almost certainly be a draw.

Lots of those to choose from. No idea.

I mean a great game by both sides, but one side still wins

Best... game... ever.

The two games that get the most votes from GMs are Botvinnik-Capablanca 1938 and Topalov-Kasparov 1999. 

Those two have routinely been voted the best in history. 

What is the deal with Fischer playing 1.c4? Did he not have his glasses on? Reached for the wrong pawn?

Spassky-Fischer, game 13, 1972

Not saying it was necessarily the most accurate game ever played. It was just a gritty streetfigting slugfest, between two chess geniuses who each refused to quit.

These forums should just be called the Fischer threads. Political sites are replete with references to a certain German dictator and contemporary Soviet dictator. This  site can't get away from Fischer. 

Believe it or not, Fischer was not involved in every great game. The greatest games, as voted on by people who know, didn't involve Fischer.

But Fischer's match game 6 against Spassky (1972) was probably his best ever game. Other great Fischer games include Fischer-Petrosian (game 7) 1971, Fischer-Panno, as Pfren mentioned, and Tal-Fischer, 1959. Oh, but Fischer was on the losing end of that game, so perhaps it shouldn't be mentioned. 

One remarkable thing about this is that Fischer himself was quite objective about the actual game of chess. He would have agreed about the quality of Botvinnik-Capablanca and Topalov-Kasparov even if he hated the people playing.

http://www.chesscentral.com/Famous_Loser_ChessGame_a/277.htm

The Immortal Game

Smyslov Fan

That's twice you've mentioned Topolov-Kasparov. Wasn't Kasparov white in that game?

Yes, Varelse. I got it reversed.

Some other Kasparov games that rank very high on the all-time lists compiled by GMs (in Informant and other sources) include Karpov-Kasparov game 16 1985 and Kasparov-Karpov game 16 1986.

Here's the Kasparov-Topalov game without notes:

Fischer's game of the century, 1956 (Fischer - Byrne)

Fischer was questionable in his morals, but man is this just such a fun game to watch through. It's always amazing to see Fischer's magic in action, how it switches from tactical beauty to strategic beauty is always so fantastic. It's a game I hope to learn by memory. It's my favorite, although I do agree that Kasparov - Topalov 1999 is probably the greatest game ever played, just considering how complex Kasparov's calculation skills are, especially the climax of that game, it truly is just so beautiful. I still think Fischer's game of the century my favorite, nonetheless.

And admittedly, I also love Fischer's own immortal against Robert Byrne. Even if it isn't as extravagant as Kasparov - Topalov or the game of the century, I've always loved Fischer's immortal, it's just a very fun game to watch through.

If you're looking for some beautiful games, I'd recommend Rotlewi - Rubinstein, it's spectacular finish is something I will never forget. There's also Zukertort - Blackburne, which is a very fun game to watch through, especially the finish. And if you really want something that's just a blast, a game that resurrects the romantic era, then Gregory Serper - Ioannis Nikolaidis is a beautiful experience, with a climactic finish in which Serper is walking on thin ice and there are few moves for him to make!

Having asked this question that you are asking, and then done hours of research looking for the most beautiful games, I must say that this post could be so much longer, but alas, I'd be typing this out for hours. Truly, Chess is such a beautiful game indeed. Now to get back to blundering my queen every game.