There really is a lot of disagreement on this topic, so I thought I'd add a few more thoughts to what artfuldodger very accurately stated above.
When asked this question, former world champion and legendary badass Alexander Alekhine replied that he can see many moves ahead. This makes sense, because Alekhine is considered one of the most powerful calculators in history. He was known for pulling off brilliant sacrificial combinations that often didn't pay off for ten or fifteen moves.
Richard Reti, however, was the grandmaster who replied with the above mentioned quote, "I only see one move ahead. The right one."
This raises a fascinating question! Is one of them lying, or do some high level chess players actually think that differently?
The simple answer is this: There are different ways of approaching positions in chess. One way (positional play) involves a deep understanding of strategy and a certain amount of intuition. Another way (tactical play) involves immediate, short-term attacks and generally requires very accurate calculation. Usually, when a position doesn't have any viable tactical resources, a good player tries to create them by making solid positional moves that improve his pieces and weaken his opponent's pieces. An example of a positional move would be relocating your knight because while it's okay where it is, experience tells you that if you can get it to a support point on the 6th rank it will probably be a game-winning advantage. You aren't seeing the win, you're seeing the creation of an imbalance that will very likely help you win. Another important trait of great positional players is that they are often able to intuitively make quick, accurate moves in positions where lesser players could spend ridiculous amounts of time (not to mention energy, which is also important in serious chess) calculating. They just sort of know. This is the power of the human mind, and it's why some strong masters I know can play five minute games that are more complicated and correct than my five hour tournament matches.
That's an oversimplification of course, but what I'm trying to get at is that the two are different yet inexorably tied together. You could say that superior positional play tends to create tactical opportunities.
Now, some players specialize. Take Jose Raul Capablanca for instance. Another world champ and probably the best natural chess player of all time, he was an amazing positional player and a defender without equal. He would play subtle positional moves and slowly dismantle opponents when they tried to attack, calmly disorganizing their forces and picking off pawns before trading down to a won endgame.
So while Capablanca and Reti were brilliant intuitive strategists, players like Alekhine were tactical masterminds. Different sides of the same coin, perhaps - all grandmaster-level chess players have to be incredibly good at both approaches, but players like Capablanca and Alekhine became world champions because they were unmatched geniuses at one of them and were able to build a style around it. This made their kung fu very strong indeed.
So there's an exploration of the quote, and hopefully I've also made the point that a master has to be able to see much further than the average amateur even if they aren't tactical geniuses.