The Affordable Care Act (ACA) seeks to cover most of the millions of Americans who lack health insurance using several methods, none bigger that its Medicaid expansion. After the expansion, nearly all Americans with income up to 133% of the federal poverty level would be eligible for government funded medical care. (Without the ACA, Medicaid eligility not only requires poverty -- extreme poverty in some cases -- but also inclusion in a category, such as disability, being above or below a certain age, etc.)
Under the Supreme Court's recent decision in National Federation of Indepedent Business v. Sebelius, the ACA's Medicaid expansion survived, but states have the opportunity to opt out of it. The federal government is funding all of the new Medicaid costs for the first two years of the expansion and at least 90% of it after that, so it's unlikely that many (if any) states will opt out (though some are threatening to do so).
Okay, so that explains the Democratic Party's view of what the country should be doing about health care for poor people. And what about the Republicans? Well, if they win the White House and Congress in November, they want to enact major cuts in Medicaid, as explained in this article by Micahel Hiltzik. Another article provides more detail.