Universal coverage" refers to a health care system where every individual has health coverage. According to the U.S. Census Bureau, there were 29 million* Americans without health insurance coverage in 2015 (this was a sharp reduction from the 46.6 million who had been uninsured a decade earlier; the reduction was due to the implementation of the Affordable Care Act).
In contrast, there are no uninsured Canadian citizens - their government-run system provides universal coverage. Thus, Canada has universal health care coverage, while the United States does not (it’s important to note that the 29 million uninsured in the U.S. includes an estimated 4.7 million undocumented immigrants. Canada’s government-run system does not provide coverage to undocumented immigrants).
On the other hand, a “single-payer system” is one in which there is one entity—usually the government— responsible for paying health care claims.
In the U.S., Medicare and the Veterans Health Administration are examples of single-payer systems.
In most cases, “universal coverage” and a “single-payer system” go hand-in-hand, because a country’s federal government is the most likely candidate to administer and pay for a health care system covering millions of people. It is very difficult to imagine a private entity like an insurance company having the resources, or even the overall inclination, to establish a nationwide health care coverage system.
However, it is technically possible to have universal coverage without having a single-payer system. For example, some experts have suggested that the United States incrementally reform the current health care system in the country to provide a government-funded safety net for the sick and the poor (sort of an expanded version of the ACA’s Medicaid expansion), while requiring those who are more fortunate health-wise and financially to purchase their own policies.
The political gridlock that has been in place over the ACA (and now the AHCA) over the last several years make it difficult to imagine such a proposal gaining enough traction to pass. But it is technically possible to construct such a system, which would provide universal coverage while also having multiple payers.
While it is theoretically possible to have a single-payer system without also having universal health coverage, it is extremely unlikely to ever occur because the single-payer in such a system would undoubtedly be the federal government. If the federal government were to adopt such a system, it would not be politically viable for them to exclude any individual citizen from health coverage.