1) MOVE YOUR HANDS. From overhead your body should be in more of an arrow shape than a "T" shape. So pull those arms back a bit. While a "T shape" with wide arms is often easier, it can place unnecessary strain on your elbows and shoulder. Doing push ups with your hands facing inward puts the most stress on your elbow and can often lead to injury, so keep those fingers facing forward. Of course there are tricep push ups, and "diamond push-ups," but we'll get there.
Keep in mind, these types of exercises are more advanced, and should only be done once your traditional push up is solid.
Plyometic push-ups involve you pushing completely off the ground at the top of the exercise, and lowering into the bottom of your push up as soon as you land. You may have even seen "clapping push-ups" which are a variation of your plyo. Remember, this is advanced!
While there are many factors involved, the chart below will show you some basic averages for men and women. These are just some indicators to see where you are, and maybe help you set a goal. We are all at different levels, and that's ok, as long as we are doing our best. Give it a glance, but don't take it too seriously. Just try and focus on imporov