*highly simplified view*
People pay tax, the Government gives money from tax to the NHS, the NHS buy services from NHS service providers.
The change here is that the NHS will be force to buy services from a competitive market that includes private providers as well as NHS providers.
What this means for a patient is that when you go see your GP and get referred for something, like an X Ray, you might be sent to a Virgin Health Care Centre, rather than your local hospital.
You won't have to pay for it because its a private company. You won't need insurance to cover it either. Its exactly the same as before, except that your GP isn't paying your local NHS hospital for the scan, they are paying the private company.
This is NOT the same as privatising the NHS. Privatising the NHS would mean removing the Government involvement and making people pay directly for their healthcare services. By definition if the NHS exists, then its not privatised. Privatising the NHS would actually mean removing it.
There are cases for and against this. As a patient you end up with larger choice of where to recieve your treatment. In areas where the NHS services are lacking private companies could fill the void and provider an over all wider spread of quality care. The negative side is that it could see funding being diverted from NHS hospitals and providers and into private firms, which could see NHS providers having to cut the services they offer, or standards fall.
But the bottom line is that this change creates a competitive market for the NHS to buy its services from. This is NOT privatisation of the NHS. The nationally funded bucket of money that is used to purchase healthcare services to deliver to the population free at the point of delivery remains, its just the 'shop' that the NHS has to buy those services from is now bigger and filled with more than just own brand products.
I'm going to keep quoting this so that the "OMG PRIVITSATION!!!11!" brigade have a chance to place a decent counter-argument, rather than the drivel that they've read on the Internet.