This article is from the Pool & Billiards FAQ, by Bob Jewett with numerous contributions by others.
Careful! There is some variation in usage of this term, so you
need to make it clear which way you are using the word.
First, a "push out" is something very different from a "push shot". At
nine ball, the first shot after the break can be played as a push out
if declared in advance, and the requirements of ball and rail contact
are waived. The incoming player can pass the shot back to the pusher.
At pool, a push shot involves a very special kind of stroke and is
played when the cue ball is frozen to the object ball -- this stroke is
a foul. (At pool it is legal to shoot towards a ball the cue ball is
frozen to, assuming no other foul, and with a normal stroke.) In a
push shot, the tip is brought slowly, slowly, very slowly up to the cue
ball until it is just touching or about to touch, and then the tip is
accelerated for the shot. Two examples:
1. A ball is frozen to the rail close to a corner pocket. The cue ball
is frozen to the object ball and straight out from the rail. The shot
is straight towards the object ball, with the tip placed on the equator
of the cue ball with lots of side away from the pocket. Once very
gentle contact of tip-to-ball is made the tip is gradually pushed
forward and the object ball sort of slips out from behind the cue ball
and goes straight into the near pocket.
2. The cue ball is on the foot spot, and an object ball is frozen as if
it had been spotted; both are on the long string. A desirable object
ball is in the jaws of one of the foot pockets. A legal way to pocket
the hung ball is to point the cue stick at a point on the foot rail
half way between the center of the rail and the target pocket, and
shoot a normal center ball stroke. An illegal push shot is to elevate
the butt of the stick to about 45 degrees, address the cue ball for
extreme follow, and shoot a gradual push shot. In this case the cue
ball will nearly ignore the object ball, and go close to the line of
aim, rather than the double "angle" of the first (legal) method.
At pool, when the cue ball is close to but not frozen to the object
ball, and the cue ball is shot straight at the object ball with a
normal stroke, usually a "double hit" occurs. This is a foul.
At snooker, you are not permitted to play the cue ball towards a ball
it is frozen to, nor to play double hits.
At carom billiards, "push shot" includes any shot where the cue ball
is close to or touching the object ball and the shot is a foul.