Soccer players should be soccer players first and athletes second. This doesn't mean that their athletic areas of fitness, speed, strength, stamina etc. are not important, because they are, but skill and specifity and perhaps spirit are the most important.
Aerobic soccer fitness determines the level at which you can take in and use oxygen to perform an activity. An activity like walking doesn't put much stress on your body and most people can cope with this aerobic activity. Aerobic activities are activities like jogging, where you can continue without getting too tired. You work at a rate which means you don't get completely fatigued or out of breath. Aearobic training will increase the level at which this fatigue takes place, and will make your heart and lungs more efficient for exercise. You will be able to run further and faster before getting tired.
Anaerobic soccer fitness determines the level at which you can work at a high intensity. This usually means short bursts of activity, where you will often be out of breath. You are working at a level where your body cannot provide enough oxygen and your muscles need to get energy from glycogen. You can only work for a short time at this level before you get too fatigued and go into something called "oxygen debt". An examples of anaerobic exercise is sprinting. Anaerobic soccer training, will make your body more efficient at using glycogen as a stored fuel and also help it deal with oxygen debt. One effect of oxygen debt is the build up of lactic acid, which is felt when your legs for example, feel a burning sensation at the end of an intense long sprint. This lactic acid needs to be removed from muscles as quickly as possible and anaerobic training helps make your muscles more efficient at coping with lactic acid and better at removing waste products from muscles.