Push ups work the chest, triceps, and shoulders. There are variations of push ups which focus on one part of the body more than others; for example, push ups with your hands close together focus more on the triceps and inner part of your chest while push ups with hands far apart accentuate the shoulders and outer chest more. Mix it up with those two, along with medium grip, feet on higher surface, etc.Before you know it you’ll be doing clapping push ups and even one handed push ups.
They build upper arm muscle, specifically the triceps (the other side of the arm to the biceps). Also they will build pectoral muscle. Push-ups do the same thing weights do, they build muscle and make you stronger. There will come a point though that push-ups will not create additional mass / strength and the only thing adding 1 more push-up will accomplish is endurance. As such, push-ups are a pretty damn good exercise for beginners and those who are happy with a body weight workout to get serious mass you will need to hit the weights.