The abaya is an outer layer and nothing else. It is a loose full-length cloak, most often black, worn over ordinary clothing when leaving the house, and it regulates the outside while leaving the inside entirely unregulated. That is a substantial part of its appeal: what a woman wears underneath is her own business, and the same garment covers a suit, a dress, a uniform or jeans.
It should be distinguished from the items it is often confused with. A headscarf is a separate garment; a face veil is another; the abaya covers the body. The bisht, the fine cloak worn by men in the Gulf on formal occasions, is a different garment altogether despite the visual resemblance. Elsewhere in the Muslim world the same function is performed by garments with their own histories: the chador in Iran, the jilbab in Indonesia and Malaysia, the burqa and the chadari in Afghanistan and parts of South Asia, each with its own cut and its own regional meaning.
Scripture does not prescribe it. The Quranic passages on dress speak of drawing garments over oneself and of modesty without describing any particular item, and the abaya is the Arabian regional answer to that instruction rather than the instruction itself. Muslim women elsewhere satisfy the same requirement with entirely different clothes, which is the ordinary state of affairs in Islamic dress.