Push ups

According to Wikipedia, a push-up (USA English), or a press-up (UK English), is "a common strength training exercise performed in a prone position, lying horizontal and face down, raising and lowering the body using the arms."

Push up is the ultimate upper body exercise. It generates strength, builds muscle, develops powerful tendons and trains the upper body pressing muscles to work in coordination with the midsection and the lower body. No other exercise in the world can achieve all these things. The bench press is often touted as a superior upper body exercise, but this is a fallacy. Not only does bench pressing isolate the upper body in an artificial way, it also destroys the rotator cuff muscles as well as irritating the elbow and wrist joints when performed over even short periods. The push up protects the joints and builds functional strength, real-world strength not just the kind of strength that can be used in a gym. This is why the push up is the number one muscle-building exercise m military training camps and academies the world over. It always has been, since the first warriors trained for strength.

A proper Push up

1. Get into starting position.

Place your hands firmly on the ground, directly under shoulders. Ground your toes into the floor to stabilize your lower half. Brace your core (tighten your abs as if preparing to take a punch), engage glutes and hamstrings, and flatten your back so your entire body is neutral and straight.

2. Lower your body.

Begin to lower your body—keeping your back flat and eyes focused about three feet in front of you to keep a neutral neck—until your chest grazes the floor. Don’t let your butt dip or stick out at any point during the move; your body should remain in a straight line from head to toe. Draw shoulder blades back and down, keeping elbows tucked close to your body (don't "T" your arms).

3. Push back up.

Keeping your core engaged, exhale as you push back to the starting position

Muscle targeted

Different forms of push up work the muscles to different degrees, but all the variations of the
push up provide great strength and muscle-building benefits. Pushups dynamically develop the network of pressing muscles around the torso, strongly working the pec major, anterior deltoid and pec minor. 

Pushups also build up all three heads of the triceps, the major muscle of the upper arm.

Technique Tips

• Avoid bizarre angles and hand positions. Find an exercise groove that's comfortable for you.
• Keep the torso, hips and legs in alignment. Sticking the butt in the air during pushups only
• occurs because the waIst is too weak to lock the body in place.
• Keep the legs together. Splaying the legs apart removes the need to stabilize the torso during
• motion and makes the exercises easier.
• The arms should be straight at the top of the movement, but don't hyper extend the elbows keep a slight kink in your elbows to prevent the joint from pinching.
• Don’t just fall to the ground – lower yourself in a slow and controlled manner. The up motion and the down motion should be at the same pace.
• Breathing should be smooth. As a rule of thumb, breathe out on the way up, and in on the way down. If breathing becomes labored and you have trouble following this formula, take extra breaths.

Speed

Clapping Push up

A lot of guys recommend doing pushups fast-maybe as fast as possible. Some even favor "plyometric" pushups-which is just the modern flashy name for the old "clapping" pushups where
you push yourself up explosively enough to be able to clap once, twice or even three times while in mid-air.
Being able to do pushups fast definitely has its benefits. Quick movements stimulate and train the nervous system by a mechanism called the "myotatic reflex." If you are competitive, many push up events have a time limit, so the faster you can do the exercise the more likely you are to

win. Besides, it's just good to know you can move your muscles fast. For these reasons, once in a while-when you are beyond beginner level and your joints and muscles are conditioned-you should do some work with higher speed pushups. However advanced you are, be sure to increase your speed gradually, over a few sessions, to allow your body to adapt.
A couple of fast sets every few workouts will be good for athleticism and variety. But despite
this fact, the majority of your pushups should be done relatively slowly; for a count of two seconds down to the bottom position, a one second pause, and two seconds back up to the top position before immediately descending again.
There are two reasons why you should try to cultivate this kind of steady pace. Firstly, smooth
technique develops higher levels of pure strength. When you move explosively, you inevitably rely
on momentum during some portion of the movement. If momentum is doing the work, it means
your muscles aren't. It's also far easier to cheat when you perform a motion quickly. We've all seen
people "bouncing" out of the bottom position of exercises, because they lack the pure muscle
power to move themselves.
Secondly, human joints adapt much better to regular movements than explosive ones. There's
less risk of chronic or acute injury. Fast movements are pretty safe to use from time to time, but
only once your joints have adapted to the regular, smoother-paced techniques. Explosive motions
can be a useful adjunct to your training if you wish, but they shouldn't become the mainstay.

Variation

Base on hand placement, elevation,... Push up has so many awesome variations which are emphasize different muscle group. You can find all of the push up variation from easiest to hardest in my article Push up progression here.