When I think of the greatest player of all time, four notable candidates come to mind. I pick these based not only on how well they played, but what their scores against other masters were, if they mastered every phase of the game or not, how successful their ideas were, and their contribution to chess theory and practice.
The candidates are: Garry Kasparov, Mikahil Botvinnik, Mikhail Tal, and Tigran Petrosian.
As you can see, I have picked two great attackers and two great positional players; to be fair to both styles.
Now, to narrow down the selection. As you know, all of these players had their individual way of playing the game, and it is hard to choose which is the overall "best" player. Indeed, one might even argue that selecting such an individual would be impossible simply because these players all had strengths and weaknesses, were all human and therefore cannot be judged by the occasional slip, and, futhermore, cannot even be judged by their wins since they have not played every single strong player in history. In addition, picking a certain player as the "best" almost forces a certain amount of bias into the decision based on the judging person's preference for style and personality. I will, however, attempt to limit my prefernces as much as possible and only bring out points about certain players that we can all agree on.
As imposing as all this might sound, there is a way to determine what player was better overall. Firstly, according to the laws of probability, if you consistently play creative moves in your games that may or may not be correct, you are sooner or later bound to come upon a brilliancy. (After millions of years, of course.) Therefore, better players have sounder styles. This does not mean that creative players are unsound; on the contrary, many creative players, Tal for instance, based their sacrifices and moves off rock-hard calculation that was not always obvious to their opponents. True, innacuracies might have been discovered later through computer analysis, but that doesn't mean that their moves were not sound. Rather, it means that the player was human and subject to mistakes.
But what play is sound, anyway?
To answer this question also involves a certain amount of bias, but I think most of us can safely say that those moves that improve our position, instigate a correct attack (One that is certain to triumph.), give us advantages, don't weaken our position, and improve the course of the game for us overall are sound. Players that consistently play these moves are sound players.
So then, of my list, which player was more sound to my mind? Based on the games I have seen from these players and the vast quantities of knowledge we have of their games, I would have to say that Tigran Petrosian strikes me as being the soundest player. His defensive technique was unmatched, he was a great tactician and positional player, and most masters considered him the hardest player to beat in history. In addition, he was the one who brought an end to Mikhail Botvinnik's long reign of World Champion when no one else could.
So we have discussed sound play, but what about master scores? As far as I know, Garry Kasparov and Mikhail Botvinnik were the two players with the best lifetime scores against master players. This particular point is open for discussion if someone can disprove this.
How about how they mastered the game? In all honesty, one of the reasons for my choosing these players was because all of them were excellent in all phases of the game!
What about playing strength? In this, I am not certain. Garry Kasparov has had the highest Elo rating to date (2851), but Mikhail Tal was also extremely strong. Very few players had plus scores against Tal, and his attacking style was usually victorious in tournament play. Perhaps if more games had been played between these two players, I would have a more definite answer.
What about contribution to theory and the game in general? Mikhail Botvinnik was hands-down the greatest contributor of these four, possibly of all time. His advances in theory are the foundation of positional play today, and his games are quoted like scriptures in strategy books. A close second is probably Mikhail Tal, with his advances of attacking play. Indeed, without tal chess would have already become purely positional long before modern players.
And what do I conclude from all this? I am rather hesitant to give my answer, especially since it will probably differ from many, many other people's opinions; and I don't wish to start any kind of argument. Suffice it to say, this is my opinion; and you can agree, or not.
It is my considered opinion (considered over the course of quite a few days) that the greatest player of all, to date, was Mikhail Tal.
He was the greatest attacking player of all time, a highly regarded author, a stupendous tactician, and a great guy to boot. His successes in match, tournament, and blitz play have nearly earned him immortality. Although some of his ideas were unsound, his concepts were so complex that most masters were simply unable to find the saving move in their games. His sacrifices, although they looked dubious in their games, were all based on calculated lines and often were not even understood by spectators until many moves later! As strong as other players have been throughout history, I have no doubt that Mikhail Tal was the greatest of them all.
As a disclaimer, the reason less modern players were excluded from my short list was because chess theory has advanced further today than ever before, and that modern players have had access to ideas and knowledge about the game that Capablanca or Lasker simply didn't have. To compare players such as Kasparov with Alekhine is simply not fair to the older masters. Their genius is appreciated today in context of their time period, and should be judged on this basis; not by modern standards.
It's a difficult question, but I'd vote for Fischer whatsoever. Although international ratings were only introduced in 1970, Chessmetrics.com uses modern algorithms to rank performances retrospectively and uniformly throughout chess history. According to the Chessmetrics calculation, Fischer's peak rating was 2895 in October 1971, much more than Kasparov's 2851. His one-year peak average was 2881, in 1971, and this is the highest of all time. His three-year peak average was 2867, from January 1971 to December 1973—the second highest ever, just behind Kasparov. Chessmetrics ranks Fischer as the #1 player in the world for a total of 109 different months, running (not consecutively) from February 1964 until July 1974. Had he played at least 10 more years than he did...