Abu Ayyub al-Ansari narrated that Allah’s Messenger (salAllahu alayhi wasalam) said:
He who fasts Ramadan, and six of Shawwal, it will be (in terms of rewards) as if the fasted a whole year. [Muslim, at-Tirmidhi, Abu Dawud, Ahmad, Ibn Majah]
So this is an established sunnah, which carries a great reward.The meaning is that the reward is like the reward of a person who is always in fast every day of his/her life.

It is highly recommended to fast six days of the month of Shawwal, but it is not obligatory. Those who want to fast can fast after `Eid Al-Fitr any six days during Shawwal.

It is not required to fast six days continuously without any interruption. One can fast according to convenience any time during the month.