What type of collision would it be when a cue stick strikes a billiards ball? Elastic, inelastic, or perfectly inelastic? Explain?

It's not perfectly inelastic, that's for sure. Perfectly inelastic is when they stick together.

Between elastic and inelastic is really how you stike the ball.

If you are dead center with ball, with the stick pointed in the final direction of motion and you follow through on the shot, then you could assume that the collision is elastic, where all the energy of the stick is transferred into the ball.

If you are off center, the stick isn't inline with the center of the ball, and you don't follow through with the shot, then the collision will not be perfect and not all the energy will be transferred into the ball. That is a inelastic collision.

The simple test for an elastic collision involves looking a what happens to all the energy. A perfectly elastic collision means that all the mechanical energy is conserved.

In the real world, the nearest you get to a perfectly elastic collision would be "non-contact" collisions, such as the collision between two charged particles which bounce apart without actually touching due to interaction of the the electric fields.

Physics experiments using gliders on an air-track often use magnets with the same pole at the end of each glider to produce an elastic collision as the magnets repel each other without the gliders actually touching each other.

In the case of your cue and billiard ball, the simple fact that you hear a 'click' as the cue hits the ball means that {a small amount} of the energy is converted into sound energy. So it's certainly not a perfectly elastic collision. Some of the energy also ends up as work done crunching up the chalk on the end of the cue.

That's why, when you analyse collisions, you consider momentum {which is conserved} rather than energy.

Inelastic, the ball is VERY solid and the cue absorbs little energy.

Nothing in reality is ever perfect though.