EMI is what you're putting into the environment, EMC is how vulnerable you are to the environment.
Edit per comment: EMI is tested by measuring what your equipment emits as conducted and radiated signals. That means putting your equipment in an anechoic chamber, hooking it up to all of its cables and accessories and pointing broadband antennae at it to pick up radiated emissions and attaching a CRO and/or spectrum analyser to all the cables (including power supply) to measure the conducted emissions.
EMC is measured by stimulating your equipment to see if it malfunctions or gets damaged. So the test starts by blasting radio at it across a range of frequencies, then putting electrical noise and glitches on its inputs and outputs, cranking the power supply voltage up and down a bit and finally by zapping any accessible contacts with a few kV to see if anything breaks.
The harshness of these tests depend on the jurisdiction doing the testing, e.g. proper CE EMC tests are more violent than FCC testing, especially with the 4kV contact and 8kV through-air discharges.