How does a galvanometer use a magnetic field to indicate the strength of an electric current?

It consists of a coil in a constant magnetic field.  The current in the coil is the current being measured, and changes the magnetic moment of the coil (given by IA where I is current and A is loop area).  This coil's magnetic moment feels a torque in this external magnetic field that moves a needle that tells the observer what the current must be, since the external field is known by the designers of the galvanometer.