Fasting six days of Shawwaal after the obligatory fast of Ramadaan is Sunnah Mustahabbah, not waajib. It is recommended for the Muslim to fast six days of Shawwaal, and in this there is great virtue and an immense reward. Whoever fasts these six days will have recorded for him a reward as if he had fasted a whole year, as was reported in a saheeh hadeeth from the Prophet (peace and blessings of Allaah be upon him).
Abu Ayyoob (may Allaah be pleased with him) reported that the Messenger of Allaah (peace and blessings of Allaah be upon him) said:
“Whoever fasts Ramadaan and follows it with six days of Shawwaal, it will be as if he fasted for a lifetime.”
(Narrated by Muslim, Abu Dawood, al-Tirmidhi, al-Nisaa’i and Ibn Maajah).
– Islam QA