Shawwal is the tenth month of the Islamic lunar calendar. It comes immediately after Ramadan. Shawwal is also the first month of the Ashhur al -hajj - (the months of Hajj). Eidul Fitr also occurs in this month. Regarding the six fasts of Shawwal Abu Ayyub (ra) reports:
The Messenger of Allah (PBUH) said,
"Whoever fasts Ramadan and follows it with six days from Shawwal it is as if they fasted the entire year." [Muslim, Abu Dawud, Tirmidhi, and Ibn Majah; Ahmad transmitted it from Jabir, Muntaqa]
Thawban (Allah be pleased with him) relates that the Messenger of Allah (blessings and peace be upon him, his family, and companions) said,
"Whoever fasts Ramadan, and then six days after Eid, it is [like fasting] an entire year. Whoever does a good deed shall have ten times its reward." [Ibn Majah, Muntaqa]
The reward of every good deed is tenfold or more. The reward of the month of Ramadhan is equal to that of ten months and that of these six voluntary fasts are equal to that of the remaining two months to complete the full year.
Although voluntary in nature, they are highly important. One can observe these six-day Saum at a stretch or with intervals, but their observance one after the other at the beginning of the month is better. Similarly, if a person has missed some Saum of Ramadan due to illness, traveling etc., he should first make up the shortfall and then observe the voluntary six-day Saum of Shawwal.