How should I test the statistical difference of a mean (or median?) concentration of an enzyme between two groups of patients where the distribution in one group is normal, whereas in other it is non-normal? The distribution of values for the entire study population is non-normal. Do I use a Mann-Whitney test, or Student's T-test?
The first thing you should do is to decide on what question is really important to answer. You say that the 2 groups have different distributions, is the difference in mean or median really the most important thing when you already know of a distributional difference?
Consider that you are looking at 2 different treatments to lower cholesterol. The first will actually raise the cholesterol in most patients, but a few will have very drastic reductions (along with medical complications either due to or resulting in the drop in cholesterol), the second treatment will result in a moderate reduction in cholesterol in all patients with some small variation between the patients. Given the above information, do you really care which has higher mean/median reduction?
If you decide that the mean or median difference really is a question of interest (and what will you do if the mean favors one group and the median favors the other group?) then there are some additional questions to ask. How non-normal is the 2nd group? (and how non-normal is the 1st group, are you really sure it is normal? or just close enough)? If the non-normality is not severe (how severe can be a function of the sample size and shape) then the t-test could be reasonable. Other approaches that do not require the normality and let you compare means or medians are permutation and bootstrap tests (but make sure that you are comfortable with the assumptions required for those). The Mann-Whitney/Wilcoxon test is actually a special case of the permutation test (but the statistic being compared is not the median (or mean) without further assumptions).