At work we can measure dust measured in micrograms using a TEOM short for tapered element oscillating microbalance.
Dust is deposited onto a tiny filter at the end of a hollow glass tube. The tube is forced to vibrate using a small electromagnet. It has a natural frequency that it will vibrate at and the machine tries to keep it at that frequency.
As the dust level increases on the filter the frequency or tone of the tube changes much like a piano tuning fork, except this tuning fork is getting heavier.
The more mass on the end of the tuning fork the lower the frequency it will vibrate. From the change in frequency over time the weight of the dust can be calculated.
This is a somewhat simplified version what actually happens in the machine, one of the interesting things is that you can use this method in zero gravity or in microgravity such as onboard the ISS.
When measuring incredibly tiny things, we use balances to find the mass. Modern day balances use magnetic force restoration. First you zero the balance. This tells the balance where the plate is without any additional mass added. You put your item you’re measuring on the plate. The balance then calculates how much mass would be required to return the plate to its original position using electromagnets. This allows us to measure very minute displacements of the plate. Meaning we can measure very tiny amounts of things 🙂
Besides the well known method of using strain gauges there is also the method called EMFR (Electro Magnetic Force Restoration) which is especially used in analytical scales to weigh really tiny amounts of material. Basically this method consists of a lever arm on which the item is placed and consequently deflects the lever arm. Then the current flowing through a coil is controlled very delicately to restore the equilibrium of the lever arm. The amount of current is used to deduce the weight. The design of an EMFR weigh cell is really ingenious when you see it the first time.
All the other answers are great; they describe practical scales that measure in the sub milligrams. But to give you something even deeper, some specialized scales can actually go to higher resolutions, measuring picograms, and more recently down to the yoctogram resolution (10^(-24) grams) which is basically the mass of a single proton.
They essentially work using a very basic law in physics. They have an oscillating beam (for the yg range, they use nanotubes for beams). Without getting too much into detail, if something is oscillating, it has a given natural frequency (a rate at which if you move it at, it will produce maximum deflection or amplitude). When you apply a force, no matter how tiny, to this oscillating structure, its natural frequency will change. This sounds insane to measure but it’s not, because measuring frequency is like counting, and we have very advanced technology for this (think atomic force microscopy). So now, if you put a mass on this oscillating beam, and you measure how much it’s natural frequency changed, you can calculate the mass of the object you added. This actually measures the inertial mass, but let’s not get into that.
Of course scales in your everyday scientific lab or kitchen or gold shop don’t use this technique, they use what other comments mentioned like magnetic methods.
Edit: English
In the kitchen… Scales use sensors that change when pressure is applied: **Piezoelectric** Effect is the ability of certain materials to generate an electric charge in response to applied mechanical stress. The word **Piezoelectric** is derived from the Greek piezein, which means to squeeze or press, and piezo, which is Greek for “push”.
Generally, small pieces of metal that bends to the weight of an object on top, called load cells, are inside.
This load cells have small “bending sensors” stick to their sides, those are called “strain gauges”
A special “amplifier” chip is commonly used to read the signal of an arrangement of those sensors and give a value to those tiny amounts of deformation of the load cell.
Then a small “computer” chip can make a relation between the deformation and the corresponding weight, even taking many measurements, and finally display the information on a screen
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