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An S-type star is a star within the S-class, a spectral class indicating zirconium monoxide spectral bands and lines. They are cool giant stars in the asymptotic giant branch or red-giant branch. They generally have similar amounts of carbon and oxygen. The presence of zirconium indicates either convection (intrinsic S-star), or previous mass transfer between binary companions (extrinsic S-star). Their effective temperature is roughly the range of M-type stars, but extrinsic S-stars can also be a bit warmer, within the range of K-type stars.
The term S star is likely sometimes used meaning such an S-type star, but is also used with a totally different meaning: a star (of any type/class) very near Sag A*, and I suspect in this case, the "S" is for Sagittarius. This group of stars is called the S-Star Cluster.