I'll try and explain...
1. 'Network Rail' owns, maintains and operates the infrastructure (tracks, stations, yards, signalling etc). This is (in all but name) a publicly owned company i.e. the UK government owns all the shares, and is the successor to Railtrack. Railtrack was a private company which went bankrupt (I was one of the many small shareholders in it).
2. Passenger train operations are run by private 'train operating companies' (TOC's) under franchise agreements i.e. the government offers a group of train services for tender, companies put in bids and eventually one is awarded the franchise. The size, length and style of these franchises varies a lot, but most of them involve some degree of subsidy from the UK taxpayer. There is a good list and map of TOC's here - http://www.nationalrail.co.uk/toc_list.html
There is also provision for 'open access' operators to apply for timetable paths to run trains on a purely commercial basis - there is only one actually doing this at present (Hull Trains), but there are other groups who are interested in this.
Most passenger stations are operated by a TOC (some of the really large stations are managed by Network Rail) but all sell tickets for other operators, and are legally required to be impartial when doing this.
3. Almost all passenger rolling stock is owned by leasing companies, and is usually 'dry' leased to the TOC's i.e. the TOC drives the trains and performs the routine maintenance, the leasing company is generally responsible for the heavy maintenance and refurbishment.
4. Freight trains are run on a commercial, 'open access' basis by various companies - EWS (mostly owned by CN) is the biggest - and freight is definitely one of the sucess stories of privatisation.
5. Timetable paths etc are agreed between the various TOC's, freight companies and Network Rail in an ongoing process (with the fallback of an independant Rail Regulator in the case of disputes), and then the train operators pay Network Rail for the use of the infrastructure. Network Rail controls the signalling system (equivalent of train despatching in the US) and hence the actual train running. There is a system of financial penalties/compensation if one TOC's trains delay another or Network Rail fouls up.
Generally the system works well (passenger numbers and freight traffic is well up compared to British Rail days) but is is costing the UK taxpayer a LOT more than before.
Hope that helps a little [:)]
Tony