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.
The Six fasts during Shawwal must be done after making up for any missed fasts in Ramadan, as the fasts during Ramadan are obligatory, where as fasts during Shawwal are Sunnah.
To gain extra reward, you could also perform your Shawwal fasts on Mondays and Thurdays of that month.
[Jami` at-Tirmidhi: Book 8, Hadith 66]