What is the name of the region around a magnet where magnetic forces can be detected?

The magnetic field is the region around a magnet where magnetic forces can be detected.

A magnetic field is a vector field, meaning it has both strength and direction at every point. Field lines are a visual representation: outside a bar magnet they are conventionally drawn from the north pole toward the south pole, while inside the magnet they continue to form closed loops. Field lines are not physical strands.

Magnetic fields can be produced by permanent magnets, electric currents, moving charged particles, and changing electric fields. Earth has a large magnetic field generated mainly by motion of conducting molten metal in its outer core. That field helps deflect many charged particles from the solar wind.

The magnetic field is commonly represented by B and measured in teslas in the SI system. A magnetic domain is only a region of aligned magnetic moments inside a material, so it is not the same as the surrounding field.