Shawwal is the 10th month of the lunar Islamic calender. Shawwal means to lift or to carry. The first day of Shawwal is Eid-ul-Fitr. Many Muslims observe the fasting for the six days of Shawwal meaning the first day after Eid-ul-Fitr. These six days along with Ramadan are equivelent to fasting for an entire year

An Important Benefit of Fasting in Shawwal:

An important benefits of fasting six days of Shawwaal is that is makes up for any shortfall in a person’s obligatory Ramadaan fasts, because no one is free of shortcomings or sins that have a negative effect on his fasting.
On the Day of Resurrection, some of his naafil deeds will be taken to make up the shortcomings in his obligatory deeds, as the Prophet (peace and blessings of Allaah be upon him) said:

“The first thing for which people will be brought to account on the Day of Resurrection will be their salaah (prayer). Our Lord, may He be glorified and exalted, will say to His angels – although He knows best – ‘Look at the salaah of My slave, whether it is complete or incomplete.’ If it is perfect, it will be recorded as perfect, and if something is lacking, He will say, ‘Look and see whether My slave did any voluntary (naafil) prayers.’ If he did some voluntary prayers, [Allaah] will say, Complete the obligatory actions of My slave from his voluntary actions.’ Then all his actions will be dealt with in a similar manner.” (Narrated by Abu Dawood).