City, river, province before the name: thành phố Hà Nội

For a named place, the word for the kind of place comes first and the name after it: thành phố Huế (the city of Huế), tỉnh Lâm Đồng (Lâm Đồng province), sông Sài Gòn (the Saigon River), đường Lê Lợi (Lê Lợi Street). English puts these words on either side; Vietnamese always puts them in front.

It works for almost any place with a name: chợ Bến Thành (Bến Thành Market), sân bay Tân Sơn Nhất (Tân Sơn Nhất airport), đảo Phú Quốc (Phú Quốc island), chùa Ngọc Hoàng (the Jade Emperor Pagoda).

In speech the type word is dropped when everyone knows the place: Em ở Đà Lạt. Keep it when the name alone could mean something else: sông Sài Gòn is the river, Sài Gòn the city. In an address it's usually there: Nhà em ở đường Nguyễn Trãi.

The city's official name is Thành phố Hồ Chí Minh, written TP.HCM, and that's what you'll see on forms and in the news. In everyday speech most people still say Sài Gòn.