To perform a standard push up with good form, place your weight on your hands and feet, with your spine and head in alignement with your legs, and your hands just slightly wider than shoulder-width apart (palm flat on the floor). Lower your upper body to the floor, flexing the elbows, then rise back to the start position. Keep your head still, and keep your eyes looking down. Breathe in on the way down, and out on the way up, and pull the abdominal muscles tight throughout the exercise.
Push-ups which are performed with your elbows in (closer to your body) put more emphasis on the triceps. Wide arm push ups with arms at a 90 degrees angle to your body focus on your chest.
Some people like to keep increasing the reps (100 continuous push ups is often seen as a good target), but if you really want to increase your strength, you should instead move through a progression of increasingly difficult exercises.
Start at whichever point of the progression is relevant to you.
Unless otherwise indicated, you should be aiming to complete 6 sets of 6 repetitions of the given exercise within a workout.
The first set is a warm up set: go back two or three levels in the progression and perform that variation for 6 reps.
Perform each following set ‘to failure’ (e.g. 6,6,5,5,4).
In each new workout, you should aim to improve on the number of reps you performed previously by at least 1, e.g.: (6,6,5,5,4) → (6,6,6,5,4)
Once you manage 6 sets of 6 reps, you are ready to move on to the next variation in the progression.
If you find yourself not progressing any longer, take a few rest days, then resume training.