I am sure this is only the first of many times I will write about single-payer health care. I will try to keep this to a brief overview.
Free-market Capitalism or Libertarianism generally support the idea that outcomes are best when companies are free to do whatever they want, and consumers are free to do whatever they want. Demand from consumers will control what companies offer and at what prices. Companies succeed that satisfy consumers the best.
Socialism generally supports the idea that consumers are best served when the government applies more control to companies to sell what the government thinks is good for the consumers.
The Current US System: We have many different private insurance companies, plus Medicare for the elderly (and ESRD), plus Medicaid for the very poor, plus S-CHIP for kids in not-rich families. The many private companies compete with each other for customers by advertising and offering different insurance packages at different prices. The US private insurance system is mostly free-market capitalist, but also has various regulations at the state and federal levels. The private insurance industry heavily lobbies congressmen and makes substantial campaign contributions to ensure that legislation is not passed that would hurt its profits. People who cannot afford insurance also cannot afford lobbyists. Politicians are not required to understand economics, ethics, or health care systems, and most don't. They often end up voting how they are told, which gives us things like Medicare Part D that explicitly prohibits Medicare from negotiating for lower drug prices.
Foreign Systems: The other industrialized nations that we usually compare the US to (European countries, Japan, Canada, Australia) are far more socialized. They can all be said to have single-payer systems, though their systems vary, because citizens have the right to most health care treatments and the government pays. The government is the single payer (a couple countries involve the government distributing funds to a handful of insurance companies based on citizen-chosen enrollment). As a single-payer, the government has great leverage that it uses to demand quality and efficiency from service providers.
Outcome comparisons: No one should ever make decisions based on belief in ideology when there is data from measured outcomes that demonstrates which system works best. The US system results in generally comparable health outcomes for the people who get health care to those in our socialized counterparts. Unfortunately, we pay about twice as much money for those comparable outcomes, and tens of millions of our citizens don't get any basic health care at all because they can't afford insurance or care. In the absence of affordable basic care, our uninsured and underinsured citizens wait until they have acute problems that cost devastating amounts of money, often causing bankruptcies and home foreclosures. Nearly half of the home foreclosures that precipitated our economic collapse were due to medical bills. This simply does not happen in single-payer systems. Money is wasted in our system on advertising and bureaucracy. If Medicare were offered to all Americans, the overhead savings would pay for the health care of our currently uninsured. With this health care, acute health problems would be reduced, and our nation's productivity and economy strengthened. America currently has the worst infant mortality rates among industrialized nations because uninsured and underinsured women cannot get effective prenatal care.
Each system does what it is designed to do. The US system is designed to make money for insurance companies that in turn give money to lobbyists and campaigns, and deny basic health care to tens of millions of people, increasing infant deaths, weakening our whole economy and bringing down quality of life. Single-payer, socialized systems are designed to give everyone adequate health care, resulting in stronger, healthier, more productive and functional nations.