We all know that push-ups are difficult, but they won't feel any easier if you forget to breathe. There are a few reasons you want to breathe during push-ups - one is so you don't get lightheaded and pass out. Another reason is because when you breathe, you're able to activate your abdominal muscles, and it can help you improve your strength.
Regardless of the push-up variation - my current favorite is the close-grip wide-grip push-up - you should inhale when lowering your body down to the ground. This will help you recruit and engage your abdominal muscles and help you keep your spine in a neutral position.
When you're ready to push your body back up to the starting position, you should exhale. I find that the exhale helps me "power" myself back up - don't forget about maintaining good form - and makes every rep I perform feel slightly easier than if I were to hold my breath. The next time you've got a few sets of push-ups in your workout, give this simple trick a shot!