Sitta Shawwal
Prophet salallahu ‘alayhi wasallam said: "Whoever fasts Ramadan and follows it with 6 days of shawwal, it will be as if he fasted for a lifetime" Sahih reported by Muslim, Abu Dawood, al-Tirmidhi, al-Nisaa’i and Ibn Maajah).
In another narration: “Whoever fasts for six days after (Eid) al-Fitr has completed the year: (whoever does a good deed (hasanah) will have ten hasanah like it).” According to another report: “Allaah has made for each hasanah ten like it, so a month is like fasting ten months, and fasting six days completes the year.” (al-Nisaa’i and Ibn Maajah. See also Saheeh al-Targheeb wa’l-Tarheeb, 1/421).
The 6 days fasting is called 'SITTA SHAWWAL'.
This bonus is available in this month (shawwal).
It is Sunnah not obligatory
It is not compulsory to fast it consecutively, you can leave gab(s) but make it up to six before the end of shawwal.
Make up your missed days during Ramadan first before 'sitta shawwal, though some jurists consented to the permissibility of Sitta Shawwal before making up missed days during Ramadan if time will not permit in the month of Shawwal but the most evidenced opinion is the former.
You can start fasting of Sitta Shawwal on the 2nd of Shawwal, if you like, but it is Haram to fast on Eid day (1st of Shawwal).
More Ref: Muslim 1164, Abu daud 2433, Ahmad 5/280
May Allah accept it from us as ‘Ibadah