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 long distance event like 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.
Beginner runners may want to start out running every other day, to give themselves sufficient recovery time while still building a running habit. You can either take a complete rest day or do a cross-training activity on your days off from running.
If you're a more experienced runner, one or two rest days should be sufficient for injury prevention and recovery. Be careful that you don't let rest be an excuse for not running . You'll need to stick to a consistent schedule of running if you want to achieve your training goals and reach your desired fitness level.