Body fat is measured several different ways.
One is underwater weighing with all air expelled from the lungs and a computer calculates your body fat%. This is very accurate but tends to cost money.
Another is through the use of electromagnetic current through the body. Also very accurate but not friendly to the wallet.
The most common and inexpensive way is through use of skin calipers. A person who has been trained to take the measurements uses calipers to measure the thickness of a person's skin folds and various different parts of the body. Then the sum of the skinfold measurements is compared to a chart to give the person's body fat%. This is still pretty accurate if done by someone who knows what they are doing (I used to have to measure body fat all the time when I was a personal trainer). It's also much easier on the wallet.