What exactly is transmit power in wireless communication?
The transmit power for a single output transmitter typically refers to the output of the transmitter power amplifier that is connected to the transmit antenna, and that power is almost entirely radiated over the air. It is power in Watts.

The power needed by the transmitter includes this and everything else to run the transmitter. The power amplifier (PA) often dominates this power requirement which is why PA efficiency is such a big deal (For example, if a PA has a 50% efficiency and is transmitting 2W, it will require 4W of DC power to run). PA's are more efficient the closer we run them into saturation, hence constant envelope modulations such as FM (which can be saturated with no further distortion) are attractive when power efficiency is of prime importance.