The crux of it is...
A Hospital Trust agree to a PFI (Private Finance Initiative) with a private investor;
The investor provides the Trust with money to build a new hospital, buy new equipment, rennovate buildings etc.
The Hospital now effectively 'rents' the building, equipment etc. from the PFI company, and pays it back over a period of time (e.g. 50 years).
That's basically it - it's just like getting a very wealthy partner, but then having to buy them out. It gets complicated and creates problems though as a PFI typically has a commercial interest of some kind though (e.g. it may be a building firm, or a firm of accountants). If the PFI was, say, McAlpine Builders, and the hospital Trust decided it wanted to build a new ward, it couldn't go to Joe Bloggs Builders down the road who'll do the job for �20,000. Instead, it has to use the PFI's choice (since it's their money that's paying for it) - which will typically be, guess who, Robert McAlpine.