"Double-clutching was common in older cars and is still common in some modern race cars. In double-clutching, you first push the clutch pedal in once to disengage the engine from the transmission. Then you release the clutch pedal and rev the engine to the "right speed." The right speed is the rpm value at which the engine should be running in the next gear. Then you push the clutch pedal in again and lock the collar into the new gear. At every gear change you have to press and release the clutch twice, hence the name "double-clutching."
Bascially, you won't need to worry about it in your modern,street legal car. Only older cars, trucks, and racing transmissions are cars in which you need to do this