I've been doing research on radio transmitters, and though I understand how the signal is generated and how data is transmitted, I still can't understand what defines a radio's transmitting power. How is a certain transmitter more powerful than the other?
Very simply, transmitter is "more powerful" if it puts out more power. It uses a higher voltage and higher current from its "more powerful" power source, and it outputs higher voltage into a given antenna impedance (typically 50 ohms).
So, a transmitter that outputs 7V into the 50 ohm antenna is putting out 1 Watt. And if the transmitter is designed to output 225V into the 50 ohm antenna, then it is outputting 1000 Watts.
There is also some amount of gain you can get from the antenna itself. Some antennas can make the ERP (Effective Radiated Power) 2x or more what the transmitter is sending up the feedline.
A radio transmitter is generally rated by the power level (in watts) that its antenna radiates into the atmosphere. The power required is produced by an electronic power amplifier which can be constructed either of vacuum tubes or solid state devices. Vacuum tubes once ruled this application until solid state devices that could handle reasonable amounts of power (tens to thousands of watts) became available. Vacuum tubes are still used when a combination of high frequency and high power are required (google traveling wave tubes for example).