To all that are saying pushups 'don't get you big/strong' (assuming that's your goal) you are mistaken. I remember being locked up in a particular commonwealth state's county prison when I was in my short lived 'badman thug' stage in life and they didn't have any weights, it was against their policy. So was making 'artificial' weights such as filling your bin with water or your pillow case with books, etc. Those things can be curled or pressed and just overall manipulated in any number of ways to simulate/get the similar effects of weight training. If you did make these items(and people did of course) it would be deemed 'contraband' and treated just as if you'd smuggled weights into the prison, and you'd get into a lot of trouble.
Yet, even with all of those restrictions on weights, they couldn't legally stop us from 'working out', and so almost everybody still managed to brick up and get big. And it was done with simply
pushups, pullups, and dips(along with other variations of those exercises that the ex military turned current crack heads in there knew and would show, etc).
Also let it be duly noted that in many forms of martial arts there is little to non existent weight usage. Maybe in MMA there is regular weight training. But in traditional martial arts there is a lot of upper body strength training accomplished through just pushups(mostly knuckle, or fingertip). And it works. Far too often I've seen those typical high school/college athletes come to the dojo on the Saturdays or during their school breaks, all cocky and ripped from head to toe from their weight training, but then cant keep up with even our juniors during strength training
which, consequently, involves a lot of pushups.
Pushups cannot be completely discounted, and should be taken into consideration with what goals you are trying to accomplish when trying to choose between doing them or using weights.