In a true single-payer system, there is only one insurer, the government. It pays for all health care, and is able to use its regulatory and market power to hold prices down and take advantage of bureaucratic efficiencies. The country that comes closest to single payer is Great Britain, with its National Health Service. You might recall that Britons are so proud of the NHS that the opening ceremony of the 2012 London Olympics included a tribute to it.
But even the British system has some private elements to it. And moving to a completely public system from our current mishmash of public and private insurance (and mostly private health providers) would require an extraordinarily costly, complex, and lengthy transition. But maybe that’s fine with you — you can argue that in the long run, that maximizes the benefits.