I'd like to add a few points to dmd's generally accurate assessment.
QRP is amateur radio transmission at a power level at or below five watts. QRP is popular in the amateur radio or ham radio community for a number of reasons. The popularity of QRP operation has remained steady throughout the history of ham radio. The reasons for its popularity have changed over time.
HISTORY. For many decades many hams operated QRP because of governmental restrictions. Many basic or trainee licenses like the old American Novice license placed tight restrictions on output power and frequency selection (i.e. no VFO or tuner, just a crystal that bound the user to one or two frequencies). Also, the great expense of frequency-variable and high power (100W or more) radios kept many operators at five or less watts.
Many hams enjoy the challenge of low-power operation for the following reasons:
CW (MORSE CODE). While CW or morse code is almost dead in commercial and military radio traffic, hams still enjoy its use. QRP is one of the last niches in the hobby where CW remains a popular and viable communications medium. Unlike voice or "phone" communications, CW requires very little power, utilizes much more compact equipment, and places less strain on the radio (radios can be set to stop transmitting and receive in between the morse pulses). CW also requires less complex antenna structures. The elimination of a morse code requirement from many national amateur radio exams has done little to stem CW's popularity among QRPers.
COST. A ham can cobble up a QRP station for very little money (radio, antenna all in for under a hundred bucks). Many hams enjoy building their own radios and antennas. The computer controlled complexity of many new radios have closed off the radio kit market. QRP rigs are generally small, use commonly-found parts, and avoid sophisticated soldering techniques such as surface mounting. If something goes wrong, it's relatively easy to troubleshoot using hobbyist websites.
LIVING IN THE 'BURBS. Many hams have had extreme difficulties setting up antennas on their own property. Those hams that have bought into planned suburban communities are often harassed for any visible antenna structure. High power transmissions can also interfere with other residents' consumer electronics. Some hams like QRP operation because it's highly portable and easy to take to a place out from under the nose of the homeowners' association. Portable operation is not limited to QRP: many hams use their cars for QRO or operation above 5W. Still, lower power operation allows for a very small radio "footprint".
OUTDOORS FUN. Hams that enjoy outdoors activities such as hiking or camping often prefer QRP. The lightweight and easy storage of the radio gear works well with backpacking. Also, antennas can be sprayed with camo and hidden within the foliage. Outdoors inclined hams will often combine a hiking and camping trip with a mountain or natural formation that provides a great vantage point for operation.