What is the name of the procedure that conducts a low level electric current throughout the body to measure the percentage of body fat?

Bioelectrical impedance monitoring.

Bear in mind that this is not considered a very accurate method. To improve the result, avoid eating and drinking for 4 hours before testing; don't exercise heavily within 12 hours; and no alcohol or coffee within 48 hours.

See also:

Eating healthy to have a healthy weight