It's difficult to give a scope of the possibilities. An autonomous 'remote control' car has been mentioned above. Other types of robots. Musical instruments. Burglar alarms.

Take a look at the exhibition section of the arduino forum: http://www.arduino.cc/cgi-bin/yabb2/YaBB.pl?board=projects Also the projects from the "Designing with Microcontrollers" class at Cornell University: http://instruct1.cit.cornell.edu/courses/ee476/FinalProjects...