The Bench Press

When you bench press the only thing you do is bench press.

The Setup:

Unlike what some people may believe the bench press is an exercise that demands your whole body.
When you lie down on the bench make sure the barbell is set at a safe height for you to unrack.
This prevents your possible loss of tension as you unrack the weight and therefore serious injury.
The bench press is the one of the most dangerous exercises you can do in the gym because without a spotter you could literally choke to death.
Nonetheless it's awesome and increases your virility by about 72.8%.

The first thing you do is to slightly hold on to the barbell as you lie down on the bench.
Now you root your feet into the floor somewhere parallel to the bench.
It does not matter whether your feet are closer to the middle of the bench or near the end of it.
Use whatever position you feel the most comfortable. In my opinion it's usually dependant on hip and thigh flexibility so work on that if you are not satisfied with the position of your feet.
I do not think it matters if your feet are planted squarely on the floor or only your toes.
What truly matters for me is the tension I create through the floor flowing into my feet traveling from my calves into my hamstrings and glutes that are in concentric contraction which is balanced by the eccentric and probably slight isometric contraction of quads and hips.
Now that the lower body is immovable on to the upper body.

Your glutes are glued to the bench. Hold on to the barbell. Lift off your upper body. Arch your lower back. Pull your shoulder blades together to create maximum tension in the upper back.
Dig your upper back into the bench. The only thing flapping around now are your arms.
The top of my head usually ends at the front of the bench.
Choose a grip you feel comfortable.
When I normally bench press my pointer wraps around the smooth ring of the barbell that means a wide grip. On the close grip bench press I position the wrinkle of my thumb directly at the conjunction of the smooth and the rough area on the barbell.
Use the grip you feel the most comfortable with and want to do.

Fill your lungs with air. Hold it. Unrack the barbell. Rest it above your body so that it can travel in a straight line down and up again.


The Bench Press:

Before you lower the barbell make sure you pull your elbows as close to your body as possible. This further helps create tension in your upper body to prevent shoulder injury among other things.
With the air in your lungs create total body tension.
Lower the barbell in a controlled manner. 
The barbell touches the chest. If you slightly bounce or hold it shortly does not matter that much. Just make sure you are in control. If you are preparing for a powerlifting/bench press meet and the rules require you to stop at the bottom you should really practice that since it takes momentum out of the movement.[Calf raises...]
Press up and in the top position get a new breath of air and repeat as many times as possible with perfect form.

Use a spotter because there is the probability of death.