The choice of species is based on which will give the best correlation to human trials.  Differences in the gut, enzyme activity, circulatory system, or other considerations make certain models more appropriate based on the dosage form, site of activity, or noxious metabolites.  For example, canines may not be good models for solid oral dosage forms because the characteristic carnivore intestine is underdeveloped compared to the omnivores, and gastric emptying rates are increased.  Also, rodents can not act as models for antibiotic drugs because the resulting alteration to their intestinal flora causes significant adverse effects.  Depending on a drugs functional groups, it may be metabolized in similar or different ways between species, which will affect both efficacy and toxicology.
Medical device studies also use this basic premise.  Most studies are performed in larger species such as dogs, pigs and sheep which allow for testing in a similar sized model as that of a human.  In addition, some species are used for similarity in specific organs or organ system physiology (swine for dermatological and coronary stent studies; goats for mammary implant studies; dogs for gastric and cancer studies; etc.).
Importantly, the regulatory guidelines of FDA, EMA, and other similar international and regional authorities usually require safety testing in at least two mammalian species, including one non-rodent species, prior to human trials authorization.