In the name of Allah, the Most Gracious the Most Merciful, all Prayers and Blessing of Allah are upon our Prophet (PBUH), his households, his Companions and all those who follow his Sunnah until the Day of Judgment.
Allah SWT has promised a great deal of Mercy, Forgiveness and Blessings for the Muslims who will practice fasting. Consequently, the fast will bring so many benefits to the Muslims.
Allah SWT says in Surat al-Baqarah:
وأن تصوموا خير لكم إن كنتم تعلمون (الآية: 184)
It is better for you to fast, if you would know (2-184).
For all of the virtues and benefits of fasting, the Prophet (PBUH) has recommended Muslims not to restrict themselves into fasting only the month of Ramadan. He has advised Muslims to perform voluntary fasting, as much as possible, in specific occasions and months, through out the whole year, to reap the benefits of the fast.
One of the recommended voluntary fasting is the fast of six days in the month of Shawwal.
The Prophet (PBUH) said in a Hadith narrated by Muslim:
من صام رمضان ثم أتبعه ستا من شوال كان كصيام الدهر.
Whoever will fast the month of Ramadan and follow it by fasting six days in Shawwal would be like fasting the whole life.
This is based on the fact that the reward of each Hasanah (good deed) is multiplied ten times.
The Prophet (PBUH) said in an authentic Hadith narrated by Ibn Majah:
من صام ستة أيام بعد الفطر كان تمام السنة، من جاء بالحسنة فله عشر أمثالها.
Whoever fasts six days after finishing the fast of Ramadan will be like fasting the whole year, as the reward of Hasanah (good deed) is multiplied ten times.
Accordingly, by the will of Allah, the fast of Ramadan will be like fasting ten months and the fast of six days in Shawwal will be like fasting sixty days (another two months), which means fasting the whole year (ten months plus two).
In addition to the great reward, fasting of those six days could make up for any deficiency or shortcoming in the fast of Ramadan. On the other hand, the fast of those six days could be performed separately or consequently during the month of Shawwal. However, this should start after the Eid day, because the fasting of that day is prohibited.
Moreover, the Muslim is requested to fast first the missed days in the fast of Ramadan. This due to the fact that fasting the six days of Shawwal has to be performed after the completion of the fast of the whole Ramadan. Missing days in Ramadan means that the fast of the whole month has not yet completed. Moreover, fasting the missed days is Fard (obligatory) and consequently should have precedence over the Nafle or voluntary fasting of the six days of Shawwal.
However, if the Muslim, in particular a female Muslim, has missed many days or even the whole month of Ramadan, in this case the fast of the six days of Shawwal could be performed first as it has to be done, strictly, in that month. Then the missed days could be done later in the year, but before the start of the following Ramadan.
We pray to Allah SWT to bestow His Mercy and Forgiveness upon us and give us the opportunity to fast those six days of Shawwal, Ameen!