We measure the magnetic field using a sensor called a magnetometer that directly measures the strength of the magnetic field (follow this link to sensors used in my research). The sensor is usually towed some distance behind the ship (to get away from the metal of the ship) at the sea surface, or the sensor may be towed from a deeptow cable so that the sensor is close to the seafloor. Magnetometers can also be attached to a deep submersible like ALVIN, or to remotely-operated vehicles like JASON or ROPOS, or to autonomous underwater vehicle like ABE.
A magnetic anomaly is the residual value we obtain when we subtract out the predicted Earth’s main magnetic field from the measured magnetic field. We make this correction because Earth's main magnetic field is generated deep inside Earth's core, however, we are interested in the magnetic properties of the volcanic rock of the seafloor.
Magnetic field also varies as function of the distance away from the magnetic material, so measurements made a long distance from the seafloor are generally weaker than those closer to the seafloor. We have to take these effects into consideration when trying to determine the magnetic effects of the seafloor lavas.
The magnetism of a material (rather than the magnetic field it creates) is called its magnetization and is measured in units of Amps per meter or A/m. Young volcanic lavas on the seafloor generally have a magnetization between 10 and 100 A/m, in contrast an ordinary laboratory bar magnet has a magnetization 200 times that of a rock.