Fresh weight = weight recorded immediately after the plant or part of a plant is harvested. For instance, the total yield of a tomato plant would be measured in fresh weight, if you want to know how much money you can get for the crop of tomatoes.
Dry weight = Weight recorded after drying the plant or part of a plant at temperatures higher than the ambient air temperature (e.g., around 65 - 100 degrees C), to drive off the water. You get a more precise measurement of biomass without the large weight fluctuations caused by changes in water content, which can occur pretty quickly depending on the moisture status of a plant when you harvest it, or after some period in storage.