Yes, solar panels do work on cloudy days. However, their efficiency can be affected based on factors like the thickness and density of the clouds. Even in overcast conditions, solar panels are still converting photons into electric currents. Clouds, fog, and shade from trees can block sunlight from reaching solar panels and reduce their power production but overcast days should not concern homeowners who want to go solar. In the same way you can get sunburnt on an overcast day, UV light penetrates cloud coverage.
Cloud coverage can provide cooler temperatures, shade, and even precipitation, especially in rainy summer months or colder winter months. Cloud coverage does not, however, stop solar energy production.