One of the most neglected types of training for the average gym goer is explosive strength. Almost everyone is taught to lift weights with a "slow and controlled movement".
However, in all sports athletes need to be explosive. When a baseball player swings a bat, when a soccer player kicks the ball, or when a boxer troughs a punch they need explosive strength.
If your goal is to get really strong, explosive speed training is one of the most important elements. Powerlifters / Olympic weight lifters do a lots of light weight explosive speed training exercises. Unlike a lot of bodybuilders and gym buffs who do mainly slow and controlled training and pump out reps until failure.
With the bench press a lot of lifters have a sticky point a few inches above their chest. If you have ever spotted someone who gets stuck in a bench press you will often find that when you just barely assist them past this sticky point they can then finish the rep on their own. With proper explosive speed training you will be able to lift the weights more forcefully and blast past your sticky point and set some new personal strength records.
Push-ups are a great exercise to develop explosive speed training for the bench press. When you do a push up you lift approximately 60% of your bodyweight. And unlike weight training exercises, you can lift more explosively by doing push ups. I am sure that you have seen people do "clap your hands" push ups before. Where they push up so fast that they actually lift their upper body off the floor and clap their hands in between each rep.
This is an awesome explosive strength building exercise. It will take some getting used to. I suggest that you start with just regular push ups and do them as quickly and explosively as you can before trying a "clap your hands" push up (you don't want to fall flat on your face).
Westside Barbell Club (one of the most famous powerlifting clubs) uses 8 sets of 3 reps using 50% of their 1 rep max for dynamic bench press training. These are not just numbers that they just randomly picked, but what they found to be the most effective for building bigger bench presses. This type of training is for developing force and speed. Not absolute strength. They do not train to failure or use heavy weights on this day.
For the push up routine you would do 8 sets of 3 reps with 1 minute rest in between each set. Each rep should be done as fast as possible. If you can, clap your hands in between each rep, this will ensure that you are using maximum speed and explosiveness. You can do this routine once a week at home on an off days from your regular gym workouts.
When doing the push ups keep your hands just a bit wider then shoulder width. Try to keep your elbows tucked in by your sides at about a 45 degree angle from your torso, don't let them flare out to the sides. This will place more stress on the triceps.
Make sure to warm up before doing this workout. Do some arm circles, and a couple slower tempo sets of push ups to warm up your chest, shoulders, and triceps. Them move right into the 8 sets of 3 reps of explosive push ups.
Give this push up routine an honest effort for a few months. It will only take about 20 minutes once a week and you will be on your way to setting some new personal strength records.