Magnetization in measured in several ways. In
the most simple way, you can measure the mass of
an object, then apply a known magnetic field and
see how much the mass changes. Through the change
in mass you can then determine the magnetic field
strength. This is an old way to go about this
You can also measure magnetic field by passing
a magnet near an electrical conductor (say, for
example, passing a magnet through a coil of copper
wire), and an electrical current will be generated
and can be measured.
In modern electronic devices, we measure the
magnetic field using a Hall effect sensor. This is
a special device that is based on the Hall effect;
the voltage of the device depends on the strength
of the magnetic field, so by measuring the voltage
you can determine the magnetic field strength.
The most sensitive probes of magnetization use
a superconducting quantum interference device, or
SQUID. These devices need to be cooled to very low
temperatures (usually liquid helium or liquid
nitrogen), and are extremely sensitive to even the
smallest changes in magnetic field.