Domestic Medium Hair is a coat description rather than a breed. It covers mixed-ancestry house cats with fur between short and long: a fuller tail, tufts between the toes and a soft undercoat, but none of the heavy ruff of a true longhair. They come in every color and pattern, from tabby and tuxedo to calico and solid black.
Because ancestry is mixed, size, personality and health follow the cat rather than a standard. Most are medium-sized, sturdy and long-lived. In a Minnesota home with forced-air heat, the undercoat is worth keeping an eye on, since brushing a few times a week keeps mats and hairballs down. The best guide to temperament is spending time with the cat, so look at each listing's age and described personality.