What is a thing called that takes measurements?

A meter (as opposed to metre).

Or a gauge.

Instrument.

1.a tool or implement, especially one for delicate or scientific work.

2.a measuring device used to gauge the level, position, speed, etc., of something, especially a motor vehicle or aircraft.

It depends on context.

Although other answers suggest generic words like meter, gauge, instrument, etc. those words are not commonly used in practice.

More typically, we refer to the specific name of the device being used to take the measurements.

For example, a device that measures length might be a ruler. Or a yardstick. Or a tape measure. Or a caliper. Depending on the size and shape of the object and how accurate we need the measurement to be. A device used to measure weight is a scale. Or a balance. To measure current through a wire, we use an ammeter. To measure resistance, we use an ohmmeter. For voltage, a voltmeter. To measure speed we use a speedometer (from inside the vehicle) or a radar gun (from outside the vehicle or object). And so on.

So, to get the best answer, you should specify exactly what you are trying to measure.