Compression test is where you pull the spark plugs, disable the ignition and fuel, install a compression gauge in one spark plug hole, prop the throttle open then crank until the gauge levels off.

This lets you know if there is good valve / ring seal.

A leak down is where you put the piston at the top with both valve closed then pump air in the spark plug hole. Leakdown gauges then tell you the % leakage, I tend just to pump in air and judge good / bad. Air will leak out of :

Oil fill ( common leakage past the piston rings and not always bad )
tail pipe ( if the exhaust valves leak, a slight Hiss is normal on high mile motors )
Throttle (Intake valve leakage, bad as this kills power )
Radiator fill ( bad head gasket / cracked head or very rarely cracked cylinder bore )