Pillaring
Pillaring most commonly referst to pressuring an opponent's shield using shffl'd aerials and shines, used by Fox and Falco. This technique is commonly used in high-level play. Pillaring is sometimes viewed as shield pressure in general. It is also commonly mistaken for Falco's vertical dair and shine combos.
How to Pillar
Against a shielding opponent, the easiest way to pillar is to shffl a dair and immediately shine. The shine is then jump cancelled into another dair, and the cycle continues. However, other aerials may be used, depending on preference. This can be mixed up with multishines and waveshines. Jabs and (for Falco) dtilts, as well as jump cancelled grabs out of shine are also popular mixups.
Pillaring's Uses
Pillaring can pressure an opponent in a shield, forcing them to roll out of it, which can be chased. It is possible for Falco's dair pillar to be shield-grabbed, but it requires very good timing. If the opponent is not good at defending against pillaring, his shield may be broken. This is rare in high-level play.
Technicalites
Pillaring can be shield grabbed or broken out of by Samus' Up-B, Bowser's Up-B, Fox doing either shine or Neutral Air, and a few other moves. The exact timing of when to break out of the pillar depends on how the opponent times his aerials and the breakout attack used.
Strategies
SHFFLing attacks against a shield leading into a pillar can put pressure on an opponent, and force them to make a mistake.