What is this object that followed me up the street?
This is a quadcopter, so called because it uses four rotors to generate lift. They are now made cheaply in China and can be purchased in most technology-related online shops. They are remotely controlled, so it is safe to assume that the owner was close, but not in plain view. It might have been the police, but it could as well be a hobbyist, taking his new toy out for a spin.

Since most units are equipped with a cheap webcam and a wireless link, I would guess he/she was sitting in front of a computer or using his/her smartphone for controlling the device and watching the camera stream.