In the USSR the term "republic" described the territorial division entity. It was used in the same meaning as the "state" in USA. There were also so called "autonomous republics", counterpart of USA term "county".
In theory the countries incorporated into USSR kept their sovereignty, so formally there was, for example the Soviet Socialist Republic of Latvia (or for that matter Ukraine or Kirghizia), a seemingly independent member of the Soviet Union. It was all a polite fiction as the USSR was a uniform centralized state ruled from Moscow. So in a way Soviet "republics" are similar to ancient Roman "provinces". Much more seldom the term "republic" was used to refer to the type of government.