As teeth move, the blood vessels in the centre of the tooth go through a temporary hyperaemia (a reaction in which the pressure rises). This shows up as increased sensitivity of the tooth's nerve. Another reason teeth become more sensitive while they're moving is that, as they shift, surfaces that used to be hidden where the tooth touched its neighbour become exposed to the normal triggers in the mouth – cold or acidic food and drink. These parts of the tooth react with increased sensitivity for a few weeks until they adapt to these triggers. Tooth sensitivity is temporary – pain in response to cold and acidic things can come and go throughout treatment. Pain when pressure is put on the teeth is most typical at the start of tooth movement (the first few months) and can also vary.
This answer is for general information. Your own case needs an examination by an orthodontist.