We'll focus on the difference between these two words, then : common and general.
Common: This means usual and ordinary. Something that is common is found often and in many places. For example, if you say that a type of bird is very common in your country, it means that there are a lot of them and that you see them very often.
General: This means concerning most people and or things. For example, you would say that it's important to have a good general education, studying a wide variety of subjects, before you decide what area you want to specialise in.