To push is to move attacking units slowly forward the enemy base, using tactics similar to fire and maneuver. Roughly half the units get into a fortified position before the covering units advance ahead, fortify, and the process repeats.
Terrans in particular are known for pushes, using specific timing to take advantage of their strong mechanical ground armies. The terran soft push in the original StarCraft uses vultures, siege tanks, and sometimes goliaths to siege and maneuver towards an enemy base. The terran hard push also uses missile turrets to provide cheap, static air defense, cloak detection, and static scouts.