Usually it means that the plant relies on natural processes like gravity or convection to cool down the reactor after it has been shutdown, rather than needing

Usually it means that the plant relies on natural processes like gravity or convection to cool down the reactor after it has been shutdown, rather than needing "active" measures like emergency generators providing electricity to run the water pumps etc. E.g. the Fukushima reactors melted down because the tsunami destroyed the emergency generators (as well as blocking the roads so they couldn't get new generators on site in time). A newer design using, say, convection to get rid of the decay heat wouldn't have this problem.