The Qur’anic evidence for covering oneself in public, or in the company of marriageable men, comes from Surat al-Nur, verse 31 and Surat al-Ahzab, verse 59. The hadith evidence is considerable, one example being the authenticated report in Sunan Abu Dawud that Aisha, Mother of the Believers, may Allah be pleased with her, narrated that Asma, [her sister] and daughter of Abu Bakr, entered upon the Prophet, Allah bless him and give him peace, wearing thin clothes, and he turned his attention from her, saying “O Asma, when a woman reaches the age of menstruation, it does not suit her that she displays her parts of body except this and this, and he pointed to her face and hands.”
As far as covering oneself in prayer is concerned, this is a matter of consensus according to the fuqaha’ (legal scholars). In order for one’s prayer to be valid, certain conditions have to be met. One of these is covering the ‘awra, or nakedness. The fuqaha’ have defined a woman’s ‘awra, generally speaking, to be her whole body, with the exception of her face, hands, and, sometimes, feet. It is also clear that the established practice, or sunna, of the wives of the Prophet, Allah bless him and give him peace, and the female companions, may Allah be pleased with them, was to cover themselves for prayer.