The six fasts of Shawwal are observed during the month of Shawwal - the month
immediately after Ramadan.
It is mustahabb to keep these fasts wherein great rewards are earned. By adding the six days of fasting during Shawwal to the fasting of Ramadan it will earn a reward of a full year's fasting, Insha Allah. These fasts may be kept consecutively or at intervals during the month of Shawwal, i.e. spread out over the month.
Abu Ayyub (r) narrates that the Prophet (s) said, "Whoever fasted the full
month of Ramadan and then follows it with the six fasts of Shawwal, is like a person who
has fasted the full year." (Targheeb)