In which cases would you say, "I am seeing" instead of "I see"?
To summarise other people's answers, there are three uses of "I am seeing":
Habitual in the present. ("I'm seeing my analyst every week now") Nearly always in transferred senses of "see" = "have an appointment with" or "have as partner", but can be used literally in eg "I'm seeing that more and more nowadays", to emphasise the ongoing process (cf 3).
Definite appointment in the future: "I'm seeing my doctor next week"; "I'm seeing Harry Potter 7 tomorrow". Again, usually in a non-core meaning of "see".
Present sensation, strongly emphasising the current and continuing process: "Yes, I'm watching him on CCTV. I'm seeing him walk up the stairs, and go into the room". Generally verbs of sensation don't take the present continuous, except in this particular highly marked construction.
It should usually be for a process in progress.
Should you send a quick message from a movie theater during the projection of a film, you would type:
"I am seeing this movie (as in "right now") and it's awesome!"
(and not "I see...")
The expression "to see someone" (meet regularly as a boyfriend or girlfriend) is more often used as:
"I'm seeing someone"
(as in "I am still engaged in this relationship right now")
"To see" is what is considered a punctual achievement verb; a verb that happens instantaneously. Other examples of achievement verbs are: catch, faint, hit, kick, recognize etc.
Many of these verbs cannot take the progressive form to describe an action that is happening at the moment of speaking. For example, we can't say,
"He is seeing the movie." or, "I am seeing myself in the mirror."
Rather, in these cases, we use the simple present tense:
"I see you!" "Do you see that crazy guy over there?"
Furthermore, using achievement verbs in the progressive tense means that the action is repeated (iterative):
"Why is he kicking the door?" or, "I'm catching butterflies!"
Now, onto the present progressive, "I'm seeing..." As stated earlier, this can be used to express
Habitual, repeating action in the present: "I'm seeing my chiropractor once a month now."
Action in the near future: "I'm seeing the doctor tomorrow."
- To convey a polite tone (colloquial, less common): "I'm just seeing if you needed any help."