One arm push up progressions from easy to difficult. Also lots of intense one arm push up variations.
One arm push up is a real strength builder that develops strength and size in the chest, shoulders and triceps. Your core also needs to work incredibly hard to stabilize the trunk (anti-rotation).
Once the basic push ups become easy, starting to work on one arm push ups is a great way to progressively increase the difficulty of your horizontal pressing exercises.
The biggest benefit of one arm push ups is that they can be done anywhere: at home, at the gym or outdoors. It's a fantastic option for people who prefer to train by using only their bodyweight and prefer to workout at home.
You can do one arm push ups on your palm, fist, bar or on fingers. The fist increases the activation of the triceps (which is why it's a lot harder) and makes the exercise easier on the wrist.
Many will point out that the correct form is to tuck the elbow. It's definitely a good form, but it's not the only correct form.
When your elbow is flared (as in the rocky push up), you will use more of your chest muscles. When the elbow is tucked (inside), you will hit more the triceps and shoulders.
Flaring elbows is not "bad for your shoulders" (unless you have pre-existing shoulder problems and in this case most normal exercises would be bad for your shoulders).
Many people actually find flaring your elbow to be much easier for the shoulders than the "correct form" where your elbow is tucked.
People think flaring your elbow is bad for your shoulder, because most people consider the tucked elbow one arm push up the correct form. In reality all forms below are the correct ones - they just emphasize different parts of the body.