The area in which the attraction and the repulsion of a magnet's poles are felt is a (n)

The are is called a magnetic field.

That's the description of an "magnetic field".

The way your question is worded, it sounds like there's an end to the field of a magnet ... it exists only up to a certain distance, and past that, it doesn't exist. This is not true. Yes, the field of a magnet is weaker as you get farther away from it, and eventually you can get so far away that you can't detect or measure it. But it's still there, and we can always calculate how strong or weak it is, at ANY distance, even in a middle-school classroom on the far side of a different galaxy from ours!