In order to activate a QTE, the enemy must first be sufficiently weakened. Upon reaching that point, a chain of actions can be started, which will severely weaken, or even kill, the enemy.
Failing to properly execute the chain will result in Kratos' attack being countered, thus getting hurt himself; in rare cases, killed. In some cases, these actions can be skipped altogether, having Kratos instead resort to attacking and killing his enemies in normal fashion.
Completing a QTE against regular enemies often grants red or blue orbs, even though exceptions exist where they grant more red orbs than usual, or in the case of the Minotaurs in the first God of War game, only green orbs.