Fasting six days in the Islamic Month of Shawwal Al-Mukarram is mentioned in a Prophetic hadith. Prophet Muhammad (peace and blessings be upon him) is reported to have said: "Whosoever fasted in Ramadan and then followed up with six fasting days of Shawwal, it is like fasting every day." (Reported by Muslim)
Abu Ayyub (may Allah be pleased with him) narrates that the Messenger of Allah (peace and blessings be upon him) said: "Whosoever observes fast in Ramadan and then follows up with six fasts in Shawwal, will be regarded as though he had been fasting every day." (Reported by Ahmad, Muslim, Abu Dawud and At-Tirmidhi)

EXPLANATION:

The meaning of above hadith 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 Al-Mukarram, but it is not obligatory. Those who want to fast can fast after Eid Al-Fitr any six days during Shawwal Al-Mukarram.
It is not required to fast six days continuously without any interruption. One can fast according to convenience any time during the month of Shawwal Al-Mukarram.