What types or kinds of chess players are there? I am often asked the question what type of chess player am I, but I am unsure how to answer this question because I do not exactly know what would define each type of player.  Interestingly, I am only asked in particular if I am an attacker or a positional player, but surely, there must be more types of players than these.  What are the types or different kinds of players? Also, if you could include a definition of each type and include any famous players as examples for types, that would be great as well.
There are two broad-brush categories of players: positional and tactical.  Positional players' styles tend to lead them to play moves that choke out their opponents.  They are patient and bide their time.  There's no better proponent of this type of play than Tigran Petrosian. He's recognized as one of the hardest players to defeat.  A more recent example would be Anatoly Karpov, another player that was just so hard to defeat. Positional players keep taking their opponents' good moves away until nothing is left but bad moves.  Aron Nimzowitch was the father of this type of play, though he did not invent it - he described and defined the style in publications such as My System. 

There are more tactical players.  These include pretty much all the darlings of chess - the ones scoring points with fireworks.  They tend towards open positions with lots of complexity, relying upon their tactical skills to force an error onto their opponents.  Examples include Rudolf Spielmann, a player from another age who made Tal look tame.  Mikhail Tal is the more modern archetype - a player able to conjure devastating attacks from seemingly nothing at all.  Then there is Kasparov, Fischer, etc the list is rather endless.

In short, tactical players do it to you.  Positional players make you do it to yourself.

At any semi-professional level of play, both styles of player can uncork combinational fu.  So this isn't to say that a positional player avoids combinations and sacrifices.  You won't normally find one player who plays both styles; it is a philosophy sort of thing.  They believe one way is best and sharpen their skills to this belief.