How many moves ahead does Anand calculate in his mind? I always wondered about this! How many moves ahead can strong players like Anand actually 'see' during the game? How does the calculation of a supercomputer like Deep Blue differ from that of strong human players?
first of all see here. here is a quote


  Asked how many moves ahead he can think, Kasparov replied that it
  depended on the positions of the pieces. "Normally, I would calculate
  three to five moves," he said. "You don't need more.... But I can go
  much deeper if it is required." For example, in a position involving
  forced moves, it's possible to look ahead as many as 12 or 14 moves,
  he noted.


for a supercomputer it obviously depends on how well-written it is, there is a concept called alpha-beta, which is used to limit thinking into unnecessary moves, like the computer usually wont consider the opponents (white) c3,b5,d6,f7, combo, on the first move. however after c3, b5 it should consider d6, and have a reply ready. from a page called Creating a chess engine from scratch on chess.com:


  A computer can easily evaluate a few million positions per second, a
  human probably 1-2 positions per second!! Usually speed is measured in
  MNodes/sekund which means million positions (Nodes in computer science
  jargon) per second. Fritz running on my old laptop does about 2.5
  MNodes while Deep Blue did about 200 MNodes per second. Raw power is
  not all - the evaluation function is also very important. Pretty much
  all engines use the same algorithm for searching through the search
  tree of possible moves to find the next move. This algorithm is known
  as alfa-beta search or it is some variant of this.


an important difference is that a computer has to consider things a human takes for granted, such as the concepts behind a known opening, even though a good program can remember which combos work (and rate them, then play based on  its ratings), its extremely difficult for it to recognize concepts, such as a fork independent of the result, this can be a advantage because a human may fork, and not see another better future move.

hope that helps!