Double declutching as the name suggests is pushing the clutch peddle down
twice to disengage the clutch plates.
One pushes the clutch peddle down, selects neutral and raises the clutch peddle.
Then one matches the engine speed to the road speed, changing up you let the revs fall, changing down you increase the revs by blipping the gas (ideally above the revs required so you take the next step as the revs are falling again).
Then depressing the clutch again select the new gear.
An extension of this is clutchless gear changing which the above is followed except one does not touch the clutch peddle.
It is easier to slip out of gear if one increases the revs slightly as one moves the gear lever.
It is important not to force the new gear, just hold the lever against the gate with firm pressure, the lever will drop in as the engine reaches the correct revs for the new gear.