Microprocessors can be regarded as programmable chips that can act like simple computers. They vary in complexity, ease of use and cost. The most basic use in a payload is to read the data coming from the GPS module about its location and convert this into a protocol that can be transmitted down the radio to the ground station. However most flight computers have additional functions such as reading sensors, saving the data to some form of storage and triggering cameras. There are a wide range of microprocessors with different strengths and weaknesses and nearly everyone prefers a different one. More information can be found on the
flight computer page. The best approach to starting with a microprocessor is to get some form of development board such as an Arduino, AVR dev board from olimex. As these chips can be purchased individually you can also make your own development board - there are many designs floating around the web.