Is it obligatory to supplicate after every salah? If yes then what are rulings on it?
Its Sunnah to say after every obligatory prayer. Its an Option for which you will get 'sawab', but if you don't you wont get 'gunnah', hence its not obligatory.
You can get more detailed answers and the fiqh (ruling) here http://islamqa.info/en/21976
Salah itself is prayer, praising, remembrance of Allah. We pray specifically in Fathiha (فاتحه), Athahiyaath sitting and sitting between prostration. Prayer after salah is optional.
In regarding to the supplicate after the Salah, I should mention that as Far as I am aware, asking Allah for the forgiveness or such matters after the saying prayers (salah) is not considered as an obligatory practice. But as a common recommended (Mustahab) Islamic rule, reciting some duas is considered as a recommended practice after the prayers.