Essentially, Part C health plans are the public Medicare supplement option.
Usually the government diagrams the choice like the above image and sort of glosses over the fact that a Part C beneficiary also has to be a member of both Parts A and B just like a senior signing up for a private supplement "Medigap" option or -- usually -- when signing up for group retiree insurance. The government apparently does this because the Medicare bureaucracy is run by left-wing political appointees and civil servants (and the Republicans are too stupid to care or even understand what is happening when they are in power to fix the error). The Democrats want people to believe that Part C Medicare Advantage is private and run by a bunch of blood sucking insurance companies so it can eventually repeal Part C, something the Democrats have been trying to do since it was signed into law -- ironically -- by President Clinton in 1997.
It is not true -- as is often stated -- that you lose Original Medicare if you sign up for Medicare Advantage.