It can be confusing to try to figure out when to use "see" and when to use "watch". The first thing to remember is that they mean nearly the same thing. So even if you use the wrong one, you'll still be able to get your meaning across. It might sound a little funny, but it's no big deal. That having been said, here are the differences:
You can "see" things without trying, but "watching" something seems a little more active. For example, if you're outside at night and you think you see something moving, you say:
Did you see that?
You couldn't use "watch" in this situation. But here's an example where you should use "watch" instead of "see":
Sometimes I love to just sit on a park bench and watch people walk by.
For movies, TV shows, plays, and sporting events, "watch" is the most common verb:
A: What do you want to do?
B: Want to watch a movie?
I don't like to just sit at home and watch TV. I have to get out and do something productive on the weekend.
However, when you're talking about going out to a theater or a movie theater, it's common to use "see". This is why the example at top uses "see a movie". Here's another example:
I'm planning on seeing Pirates of the Caribbean 4 the first day it comes out.
"Watch" is still OK for describing this activity, but it's less common.
It's also more common to use "saw" or "seen" when you're describing movies or TV shows that you've experienced in the past:
Have you seen Toy Story 3?
I haven't seen it. Is it any good?
"Watch" is not incorrect in this situation; it just emphasizes the actual action of watching:
I watched Toy Story 3 with my kids last night. They were into it at first, but then one of them fell asleep halfway through.