Contributed by weezil:
Total Dissolved Salts meters are essentially little voltmeters that look at the voltage produced by a sensor, usually a couple of metal pins. The nute solution acts like a battery electrolyte and the pins function as do plates (electrodes) in a battery. The idea is that a nutrient solution is more electrically conductive when there are more nutrient salts in solution, so more salts means more voltage. A little math is done in the machine to convert the voltage to ppm (parts per million of dissolved solids).
There is a calibration adjustment so this math can be touched up to compensate for various factors. You will need a test solution to verify your meter once a week. Usually you will find a single measurement at about 1500-1700ppm is enough to verify it's reading what it's supposed to.
You need one that will read at least 0-2000ppm (or 0-1999ppm). You could use a 0-999ppm meter in a pinch if you added an equal volume of plain water to a sample from your tank-- you'd just double the meter reading.
It's best to simply get the correct meter.
There are other scales of measurement of nutrient concentration. In Europe, the "EC" (electrical conductivity) meters are preferred. They measure in units of millisiemens or mS instead of parts per million (ppm). The numbers are convertible one scale to the other, but most references and discussion here cite the ppm scale.
Waterproof meters are both more expensive and worth it.