How is hay tested?

That all depends on what you're testing it for: Are you testing it for moisture, internal temperature or for its feed value?

The former two simply involve sticking a long (around 3 ft long) probe with a digital reader attached that usually enables you to see both moisture and temperature measurements into the bale, hitting right into the core (centre), then waiting for the meter to give you the results.

The latter, however, involves taking a sample, usually from as close to the middle of the bale as possible (never take from the outer layer as that will skew the results), and sending it, by mail, to a lab specially designed to test such feeds. An example is the Cumberland Valley Analytical Services; the techs working here will test your hay based on what you want your hay tested for. The methods for testing for various nutrients are lengthy and complex, too long to mention in this answer. It is highly recommend you visit the related link below to see the various types of tests that can be run for your hay sample and how they are achieved for yourself.