How does Universal Heath care work?

Government-subsidized Insurance

Some countries make it mandatory that every citizen has health insurance and then pay for a part of that insurance. Sometimes, insurance is totally state-payed.
The country's healthcare providers are mostly private.

By allowing citizens to choose their own hospitals and clinics, it is hoped that the providers of the service will get more efficient over time, in order to attract more clients, due to cheaper prices and better service.

Single-payer System

In this strategy, Government is the only buyer of medical services, which are then resold/ distributed for free to the entire population.
The country's health infrastructure is a mix of private and public.

Instead of incentivizing more efficiency from hospitals and clinics, the Single-payer System tries to minimize costs by leveraging the fact that there is only one buyer of Health services, which forces suppliers to either accept the Governments' terms or only be able to sell to a few private businesses.

National Health Service

Finally, countries can adopt a National Health Service, both owned and paid by the Government. This allows everyone to access healthier for 'free' or at very low prices.
Most health-related businesses are public.

By building and managing one giant network instead of many small ones, the country can benefit from Economies of Scale, without having to worry about Monopolies, because the people who are ultimately responsible for the Service are politicians, who have to please their voters.

Conclusion

As you could see, UHC is not a simple thing, there are many ways to go about implementing it, and all of them can be more or less effective depending on how good or bad the execution is!