The Suppression meter generally shows you how much fire the unit is currently receiving. The more suppression, the less the pixeltrooper will return fire and observe (i.e. they will cower more the more fire is incoming). With enough supression, they may become pinned, which means they will stay put and ignore movement orders. With further degradation, they will become shaken, which means they won't respond to move OR fire commands. So suppression has the immediate effect of reducing a units output of fire, and eventually it will cause them to be pinned, shaken, panicked, etc., especially if they start taking casualties. The quality of the unit determines how effective suppression is against and how fast morale degrades in the face of continuing suppression.
A unit is "being suppressed" when its suppression meter is lit up. Suppression is a an effect of enemy fire that lasts as long as the fire continues. Pinning is a morale state that endures slightly longer depending on the unit's quality, communication with its leader, etc.