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The push-up requires many muscle groups, but the primary muscle groups are the chest muscles, the pectoralis major and minor. These are the two large chest muscles and mainly the pushing muscle group of the upper body. When pushing and lowering the body during a push up, the pectoralis major is doing most of the work.
Here is a graphic of the muscles used in push ups. Compare it with the following about bench press. Both will give you a bigger chest and some more arm and shoulder size. Push ups work the stomach more. I've been working out in the gym for more than 40 years. My advice to anyone who will listen is to do a lot of push ups.