There are a number of ways to sense electric current — and all of them involve some fuss, at least in comparison to sensing voltages. The chief reason for this is that because current is a flow, rather than a potential (like voltage) we need to measure it in series with the circuit, rather than in parallel.
For beginners, this is the reason that your multimeter behaves so much differently when you switch it into the current ranges. The resistance of the meter is now very low, as it is meant to be inserted in series with a circuit. Consequently if you inadvertently touch a voltage source with the probes, it is like applying a short circuit to the voltage, and the protective fuse in the meter blows out, something that is easy to do by the way, and that I’ve done at least once with all of my meters.
Another popular way of sensing current, is to insert a low value resistor in series with a circuit and measure the voltage drop across the resistor. This is convenient when using small DC circuits, less so when using mains circuits (120 volts or 240 in Europe). By the way, this is the way your meter senses current, and the fuse is protecting the low-value resistor, which would otherwise be liable to a smokey death in inadvertent current-measuring mistakes.
When using mains circuits, often a transformer is employed. Even the electrician’s clamp meters work this way – although the principle may not be that clear. Only one conductor is inserted in the clamp meter, which acts as a one-turn primary and the meter forms the secondary winding of the transformer. Note that if both conductors are inserted into the clamp – say a lamp cord, that no reading is possible, because the two wires have equal but opposite fields, which promptly cancel each other.
Many other small current transformers are available, but for the hobbyist, the problem of tying the circuit to the mains, then finding an enclosure to house the transformer, is also some fuss. Lately hall effect current-measuring chips have become available, for measuring current, but the fuss is still in connecting them to the circuit and housing them.
Wouldn’t it be nice if you could measure current without having to mess with cords or plugs at all? There are small hall effect sensors with linear output that can be used to measure magnetic fields. I thought that it might be possible to use these to measure the current in common electrical wires such as lamp cords. You might be quite right to wonder about the equal but opposite fields from the lamp cord’s other conductor. The trick I have used is to get the sensor physically closer to one conductor than to the other conductor, so the fields are “sensible”.