It usually involves using a small resistor (< 1 ohm) in series with the load. The resistor (0.1 to 0.5 ohms) is chosen such that it develops a small drop across it (100mV to 1V depending on the load voltage and sensitivity required). The drop across the resistor is directly proportional to the current through it (and therefore the current through the load), and can be used to determine how much current is flowing. For example, if the sensing resistor is 0.1 ohms, and develops a 100mV drop, it means that the load is drawing a current of 100mV/0.1 ohms = 1000mA or 1A.
You can connect a voltmeter (calibrated in mA or A depending on the expected max load current and sense resistor value) across the sense resistor to get a direct readout of the current. Or you could connect this to a protection circuit that trips the load when the current exceeds a certain level and so on.