Our prophet (pbuh) reminded us to follow up Ramadan with 6 fasts of shawwal. There is a reward for these fasts and it teaches us the attitude that worship never stops. If we fast these sx days we will get the rewards of fasting the whole year. The reason for this is that every good deed is rewarded 10 times. Shawwal is the only month when we are recommended to fast 6 days , for every other month it is recommended to fast 3 days.
If you have missed fasts in Ramadan you should make these days up first. If we have some leftover fasts we need to make them up in Shawwal, if they cannot fast in Shawwal then they have the full year until the following Ramadan.