Mehadrin (from hiddur, meaning "beautified" or "embellished") is the generic term describing a certain level of kashrut, as described here, covering e.g., the presence of a supervisor in a kitchen, the type of meat being served (glatt vs. non-glatt), how shmita is being handled, etc.
Badatz (acronym of Beth Din Tsedek) is a name for a kashrut organization that supervises kosher products. They typically only supervise mehadrin products, this is the case for instance of the most well known Israeli Badatz Eidat Hacharedi, Badatz Chassam Sofer from Bnei Brak or Badatz Beit Yosef.
There are other, non-badatz, agencies certifying and requiring mehadrin products, e.g., R Landau in Bnei Brak.
Regular kosher is not mehadrin, by definition. I do not know of a Badatz in Israel who would certify non-mehadrin foods.
Some of the differences between what regular kosher agencies accept and mehadrin don't are halav Israel, shmita heter mechira and no requirement of mashgiach tmidi.