I've heard of an MBA (Master of Business Administration) and MPH (Master of Public Health), but I've never heard of an MPA, so I'm not sure what they do.
I know at Suny Upstate's psychiatry residency there's the option to receive your MPA during that time. Basically the idea is for one to take on leadership roles in the government and such.
My mother did that, except she got her medical degree elsewhere (MBBS rather than MD). Got her MPA in the US, went back to run hospitals. Long hours, good pay.