Sunday is my rest day, it follows my long weekly run on Saturdays (currently 15 miles)......and I must admit I do tend to eat a fair bit in order to refuel and recharge the legs.
Mondays and Thursdays are my quality days and often involve punishing hill reps, quick 5 mile tempo runs or speedy reps work. I often follow these days with what are known as recovery runs. In my case they'll be between 8 & 8:30 pace......and their purpose is simply to stimulate the aerobic system, keep the heart rate in the correct zone and get blood to the muscles to help with recovery.
Rest days and easy days are ridiculously important for repair and eventual improvement. One or two rest weeks a year can really help recharge the mental batteries too.
The only issue for an athlete wanting to reach their potential is that if you take too many rest days a week, it limits the amount of mileage you can do. There does seem to be a correlation between increasing weekly miles and resulting competence as you get close to your potential. Much as I'd like to get away with just 2 quality days and 1 long run weekly, I would probably hit a plateau and race times would stagnate. Those recovery runs and slow easy runs really help keep the cardio/aerobic system in check. They also help you keep your weight down!
Newcomers shouldn't start with 6 training days a week though. People need to build up over a number of weeks and months.