You can do diamond, dive-bombers, planche, one-handed, incline, decline and etc. The types of push-ups you do seem pretty much only limited by the human imagination.
That being said the vast majority and variety of such exercises at times can make scientific info on them a little harder to track down. Some of the benefits seem more anecdotal than supported by science at times.
However, purported benefits that can be derived from them can be stronger glutes, abdominal, shoulders, pecs, lats to some degree and triceps. The main focus on what muscles are recruited can be increased based on leverages, weight, hand spacing and etc.
For better recruitment of the abdominal and glutes try tying a resistance band around your legs and then force your legs apart while simultaneously doing push ups. You will find that you improve your form by getting better (almost automatic glute contraction) by doing this. This is just one of the ways to modify the exercise.
Bring the palms together and you will place more stress on your triceps. Face the elbows outwards (similar to a planche) and yet again you are getting some sort of more focused tricep recruiting.
Of course these are all anecdotal accounts, but they are techniques that have been used over and over again for training gymnasts and calisthenics enthusiasts.
Injured doing the bench press or other heavy pushing movements? Simple modifications could be useful to rehab or lightly work the injured area.
From Marks Daily Apple:
If you have bad shoulders, wrists or elbows, in time your joints can be restored with lower intensity variants like the wall push-up. The body can only be as strong as its weakest link, and connective tissue tends to be slower to adapt than muscle. A novice or an injured person should start with the wall push-up, working to 20 and eventually 50 consecutive reps in each set before moving on. To perform this variant, simply lean against a wall with your toes a few feet away and do the push-up movement from this semi-upright position.
However, like I said it is hard for me to find the science on any of this. There isn't much out there that I've searched that will tell you how your growth hormones or test is affected by doing push ups. These are reports by guys who are doing the work in the trenches. People who train others and train themselves using these movements.
I know this is a weak article simply because the sources just aren't there, but I'm going to keep hunting and keep putting out information about the push up. Enjoy