After fasting for 30 days in Ramadan, Muslim then celebrate Eid ul-Fitr. Its on first day of the month of Shawwal.

In Shawwal month there is a recommended fasting that we can observe.

Abu Ayyub Al-Ansaree (Allah be pleased with him) narated that the Messenger of Allah (blessings and peace be upon him, his family, and companions) 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]

It is recommended for the Muslim to fast six days of Shawwaal, and in this there is great virtue and an immense reward. Whoever fasts these six days will have recorded for him a reward as if he had fasted a whole year, as was reported in above hadith.

Therefore, the Muslims should take this opportunity of acquiring such an enormous reward from Allah.

Why Ramadan fasting + 6 days of Shawwal fasting = One year fasting?

Some jurists (`Ulama) explained how according to above hadith, a Muslim who fasts during Ramadan every year and follows it with six days fast of Shawwal, will be credited for fasting a whole lifetime. The Jurists correctly said: a good deed (hasanah) is rewarded a minimum of ten times its equivalent. It follows, then, that one Ramadan is equivalent to ten months of fasting, and the clincher, six days, is equal to two months, (6x10=60). That undoubtedly completes the year's twelve months. Thus, we see the wisdom and the reason why the Prophet (saas) mentioned six days after Ramadan in Shawwal, not five or seven.

Shawwal fasting, Consecutive or Any Day on month of Shawwal?

The six days of Shawwal fast can, therefore, begin right after `Eid ul-Fitr, i.e., on the second day of Shawwal. As a matter of fact, that is the position of many scholars. It is more preferable to start these fasts from the 2nd of Shawwal and keep fasting up to the 7th of it. However, if, they are kept in other days, it is hoped that the requirement of the above hadith may also be fulfilled.

Sheikh Ahmad Kutty explain about this:

Scholars differ on this issue and there are two predominant views. One group states that the six optional fasts should start on the second day of Shawwal, the day after Eid al-Fitr, and the second group states that there is flexibility and one can fast the six before the expiry of the month of Shawwal. Since there is no text explicitly stipulating that the fasts must be done consecutively after Eid al-Fitr, it is an interpretation on the part of those who insist the fasts must be consecutive. Therefore, one cannot make such a hard and fast rule absolute. This is one of the areas where differences of opinion are a sign of mercy and indicate latitude within Shari’ah laws.

References: islam.ca and other islamic sites