All the three separation techniques (cyclonic, gravity and centrifuging) are most commonly used for separating out solids from other solids or liquids or gases. All the techniques make use of differences in the gravity (or density) of different components of a system.
Simple gravity separation makes use of difference in the gravity of various constituents. No other forces may be applied for gravity separation. Think about sand mixed with water, when left undisturbed, settles at the bottom of the container since it is heavier than water. This is simple sedimentation, a type of gravity separation technique. A very commonly used method in our day to day life.
Centrifugal separation also makes use of gravity differences, but speeds up the process of separation by application of centrifugal force. This is accomplished by a mechanical equipment called a centrifuge, where the suspension is rotated around a fixed axis. This causes generation of centrifugal forces, due to which the denser medium will collect either on the sides of the centrifuging containers or at the bottom. commonly used technique in industry, esp. medical and chemical research.
The cyclonic separation process makes use of high speed wind flow in a helical pattern to separate out particles from a fluid stream. Unlike a centrifuge, there are no moving parts in this separation technique. It is faster than both plain gravity and centrifugal separation technique. It is most commonly used in industrial settings, a common example is for cleaning of exhaust gases from thermal power plants.
Hope this helps.