Most runners need at least one, even two, days off a week from running. Research has shown that taking at least one day off a week reduces the frequency of overuse injuries. If you take at least one day off, your body will have a chance to recover and repair itself. You’ll find that you’ll actually feel better during your runs.
The best days for rest will depend on what type of runner you are and if you’re training for a specific event. If you tend to run a lot of miles on the weekends, then Monday might be a good rest day for you. If you’re training for a marathon and you do your long runs on Saturday, you may want to rest on Friday, so you have fresh legs for your long run.