A Defra Exempt stove has passed a lab test where its emissions are measured at its rated output and at a reduced rate. This is to help reduce air pollution in towns and cities that are Smoke Control Areas. A Defra Exempt stove can generally be used to burn wood in a Smoke Control Area - it actually depends what fuel the lab test was carried out with but this is nearly always firewood.
It is worth noting that the Ecodesign legislation which is due to come into force in 2022 sets what are arguably even stricter emissions limits than those set for Defra Exemption, but the framework doesn't include the low output test. We may well see the introduction of an additional low output test for stoves that can be used in Smoke Control Areas after 2022.