Push-ups are arguably the most popular bodyweight exercise ever; and certainly the most common bodyweight chest exercise.
THE BENEFITS OF PUSH-UPS:
- BUILDS STRENGTH: You will be lifting around 50% (knee push-ups) to 70% (toe push-ups) of your bodyweight during a proper push-up [1]. Push-ups are an efficient way to build your strength for your upper body muscles, especially your chest (pectoralis major), triceps (triceps brachii) and your shoulders (anterior deltoid). They can also train your trapezius and abs if you do them in strict form.
This diagram is taken from Bodyweight Strength Training Anatomy
- THEY ARE AN EASY AND CHEAP WAY TO TRAIN: You can do push-ups practically anywhere, as all you need is your body and some space on the ground.
- THEY ARE EASY TO TRACK PROGRESS: It is quantifiable, meaning you can see how strong you’re getting by the number of push-up repetitions you can perform each time.
STANDARD PUSH-UP TECHNIQUE:
1. HANDS POSITION: Place your hands slightly wider than shoulder width. 
2. FEET POSITION: Keep feet close together on the ground. To make them easier to do, you can do KNEE PUSH-UPS. Keep the knees together and position your bodyweight so that the majority of the pressure is just above the patella (knee cap) (see below pic).
3. BODY POSITION: Keep your core tight, straighten your legs, your body should be in a straight line from heel to head. Don’t over-arch your lower back as it can lead to strain or even injury.
4. HEAD POSITION: Make sure your head and neck are aligned with your spine. 
5. ELBOWS POSITION: With the arms at a 45-degree angle from the body, the hands should br positioned directly under the elbows.
6. BOTTOM POSITION: With the glutes (butt) and abs contracted, keep the entire body tight, lower yourself until your chest touches the ground.
7. DRIVE UP: Reverse the movement and raise your body until you are back to the start position. Avoid locking the joints while performing the movement.
[1] https://caloriebee.com/workout-routines/How-Much-Weight-Do-You-Actually-Push-Up-During-a-Pushup