They are usually called binary stars. When two stars come close enough to each other, like our Earth and Moon among the planets, they orbit directly around their own center of mass.
If our Earth and Moon were further apart in the inner part of our Solar System, they would orbit independently and both would qualify as planets (the Moon would be massive enough to "clear a neighborhood"). Instead the Earth and Moon are close enough for the gravity between them to hold them together. In this sense the Earth+Moon is like a binary or twin planet, and models of the inner part of the Solar System often treat the pair as such.
Likewise for many, even most stars. "Alpha Centauri", the apparent second closest star to our Sun, is two stars close enough to each other to orbit their own center of mass like our Earth and Moon. One difference is that in the Alpha Centauri pair the center of mass is outside both stars in the space between them, whereas the Earth-Moon center of mass is (currently) just inside the Earth.