“In hyperbolic geometry, a horocycle, sometimes called an oricycle, oricircle, or limit circle, is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional case of a horosphere.”
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Hyperbolical geometry describes a weird but possible space that cannot quite be ELI5’d but in that space directions are more “tightly packed” than usual: if you start drawing two parallel straight lines, they will diverge at a distance; area of a square is greater than pi*R^(2) etc. Turns out, in that space there’s a special kind of circles possible: they curve infinitely and have infinite length without ever reaching the starting point, their center lies infinitely far away. That’s what is called a horocycle.
I recommend trying a game called Hyperrogue to get a feel for hyperbolic geometry (it also has a TVTropes page). It can be downloaded for free on its website, and it specifically illustrates weird hyperbolic things like horocycles, some areas are based on them.
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