Pushups is a great upper body movement that involves your pectoral, deltoids and triceps. Its a great exercise for beginnersto build up starting strength and this way you can build some muscle too.
Over time if you can comfortably do more than 15 reps of few sets your body will stop building more muscle and you need to progressively overload your muscle with more weights.
The prime factor that causes muscle hypertrophy (a fancy term for increase in size of skeletal muscle) is to create muscular tension. And the way to create tension is to perform multiple sets for a muscle group in the rep range of 5 to 7.
Eventually, you need to move into more advanced movement like squats, deadlifts, military press, pull-ups and dips (both weighed). Just these five movements will cover your entire body and ultimately doing push-ups will be a mere warmup for you:)