Should I send reminder to a professor in a school I applied to, for an introduction email I sent two weeks ago?
As a busy professor myself, I personally appreciate getting a reminder email. I fully intend to reply to every email I get from a prospective student, even if only to say "Sorry, your research interests do not seem to be a good match for the work we are doing in my group." But in practice, life gets busy. I'm traveling. Or I'm booked solid with meetings and so I don't have time to immediate answer the 50 or so daily emails requiring responses. And so some emails fall through the cracks. Thus I appreciate receiving a single reminder email. Such emails help me do something I want to do: namely, be the kind of person who gives every serious inquiry the courtesy of a response. (A second reminder email is not necessary. If I haven't replied the first two times, there is likely a reason.) 

As for what to write, anything short and sweet and non-accusatory should be fine. Personally, I prefer the polite fiction that perhaps I did not receive the original email. You know it's not true, I know you know it's not true, but in phrasing your note this way we agree to mutually overlook these facts. I'd suggest something very simple, such as the following.


  Dear Prof. X, 
  
  I wrote you a letter a few weeks ago, but given the uncertainties of
  email I do not know whether you received it. Thus I am resending my
  note just in case. 
  
  With my best regards, Joe
  
  [and then append the earlier letter here]