Six Fasts of Shawwal
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,
"He who observes As-Saum (the fasts) in the month of Ramadan, and also observes As-Saum for six days in the month of Shawwal, it is as if he has observed As-Saum for the whole year.''
According to the formula that there is at least ten-fold reward for every virtue, the Saum of one month (Ramadan) are equivalent to Saum of ten months. If one also observes six days of Saum of Shawwal, they will be equal to Saum of two months. Thus, one becomes eligible for the reward of one full years' Saum. He who makes it a permanent feature is like one who observes Saum throughout his life. He will be considered by Allah as one who observes Saum permanently. 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.
The above from www.jucanada.org
The best days to fast the six are on Mondays and Thursdays and the 13th, 14th, and 15th of Shawwal.