Generalized circle
A circle or a line -- the two kinds of set every Möbius transformation sends to another one.
A generalized circle (sometimes called a circline) is either an ordinary circle or a straight line in the complex plane, treated as two cases of one family. A line is naturally viewed as a circle of infinite radius, or as a circle passing through the point at infinity on the Riemann sphere. Bundling the two together matters because every Möbius transformation sends generalized circles to generalized circles -- a line can become a circle and a circle can become a line -- and which one happens is decided by whether the image passes through the transformation's pole.
Examples
Inversion w=1/z sends the line \operatorname{Re}(z)=1 (which misses the origin) to a circle through the origin.
Inversion sends any circle through the origin to a line, since the one point that would map to \infty lies on that circle.