Put simply, the longer the lever arm, the less leverage you have, and the shorter the lever arm, the more leverage you have. When applying this to the push up, in order to maximize your ability to create force and get better leverage on each rep, keep your elbows closer to your body instead of flaring them out at a 90-degree angle, which is typical for how most people do push ups.
When done well, the push up strengthens the entire upper body pushing musculature, including the chest, shoulders, and triceps. However, many people allow their elbows to move past their wrists, either behind or out to the side of the wrists. This not only places unwanted stress through the elbow joint (which elevates risk of an overuse injury at the elbows), it also makes the push up less effective because it reduces the chest and shoulder involvement, and makes it more of a triceps-dominant movement.
If your hands are pointed slightly inward, as is often the case in how many people perform push ups, it usually encourages people to flare their arms out away from their sides, which is the issue we covered in mistake number one. Not to mention, pointing your hands inward also forces for your elbows to move out beyond your wrists in the manner we just addressed in mistake number two. In short, better hand placement encourages better elbow alignment and shoulder positioning. Turn your hands outward slightly, pointing your fingers outward, away from the middle of your body at roughly a 45-degree angle. Doing this will help keep your elbows and arm in a better position for maximizing strength and minimizing unwanted joint stress.