Here are some simple rules that will help deciding which word to use:

• See is used as inactive word; you just see without any effort:

  • you have visual impression: "I can see my home over there", "I see trees of green"
  • you understand: "I see what you mean"

• Look is used as active word, you make an effort to see:

  • you try to see: "look at this!" (maybe you have to turn your head or stand up)
  • you pay an attention: "to look for a baby"
  • you search for something: "you can look up the word in the dictionary"

• Watch is also an active word; you also make an effort, but it is for a longer period of time:

  • "I'm watching "Star Trek", "I like to go to a zoo and watch tigers playing"
  • pay attention: Watch the kittens as they may run away.

Here's an example demonstrating the difference:

I'm looking, but I don't see it.

Back to examples:
I'm watching "Star Trek" — it is a continuous action;
Have you seen "Star Trek"? — it is a question if you ever seen it at all, or do you know about it;
It can be also formulated: Have you watched "Star Trek" all night?, and it will mean that you have spent all night watching a movie.

See also: (1), (2).