In 1744, Lord Nguyễn Phúc Khoát, ruler of southern Vietnam, decreed that members of his court, men and women alike, would all wear trousers and a long, buttoned gown. The famous writer Lê Quý Đôn referred to these as an áo dài, literally, a long garment1.
The costume evolved over time, and was not present in its modern form until painter and fashion designer Nguyễn Cát Tường (AKA Le Mur2) updated the look in the late 1920s and early 1930s, resulting in the popular áo dài Le Mur, which featured a Western collar and puffed sleeves. In 1934 painter Lê Phổ created the áo dài Le Mur, in a more tight-fitting but otherwise more traditional style.
The áo dài met with the same vicissitudes of fashion as any other garment, with a not-too-popular male version making an appearance in 1952, non-traditional fabric patterns appearing in the the late 50s, and the short áo dài mini appearing in the 1960s (knee length dress, but still worn over full-length trousers). After the Fall of Saigon (1975), the new government banned the áo dài, but this ban was eventually lifted in the mid-80s, and it quickly became popular again, being held up as a symbol of traditional Vietnamese culture.
Today it remains quite popular, and is seen as a very traditional and somewhat formal dress, as might be worn to the office and especially to formal events. These days the áo dài is solely a female garment, and if you see a male wearing something that looks, to Western eyes, like a áo dài, it is most likely the closely related áo gấm.
Áo dài is pronounced /ʔǎːw zâːj/ in the North and /ʔǎːw jâːj/ in the South. Westerners usually just go with /aʊ daɪ/ ("ow-die"). The plural is 'áo dàis'. In Hán Nôm it is written 襖