This circuit is a simple battery-powered switching regulator provides 5V out from a 9V source with 80% efficiency and 50-mA output capability. When Q1 is oon, its collector voltage rises, forcing current trhough the iinductor. The output voltage rises, causing A1`s output to rise. Q1 cutts off and the output drops low enough for A1 to turn Q1.
The 1 uF capacitor ensures low battery impedance at high frequencies, preventing sag during switching. See schematic diagram below :