Category A: this is for vehicles which are so badly damaged that they are fit for scrapping only. They can never again be legally driven on a UK road and no parts can be salvaged from them for reuse, so these must also be destroyed.
Category B: Cat B vehicles are very badly damaged but some of the spare parts on them can be salvaged. However, the body itself must be crushed and scrapped so that the vehicle can never again be driven within the UK.
Category C: This is for those cars which are so badly damaged that they would cost more to repair than they were worth. However, sometimes these can be repaired if the owner is prepared to accept cheaper (including second-hand) parts or use cheaper labour. If the vehicle is repaired to a roadworthy standard it may be driven again; however repairs must be carried out professionally.
Category D: This is for a car which has been lightly damaged and which would normally be economicaly repairable. However, when other costs such as those detailed above are taken into consideration the insurance company may decide to declare it a write-off.
Category S: Cat S signifies structural damage and it has been brought out to replace the Cat C classification. The vehicle can be repaired and put back on the road, provided that the repairs are carried out to a sufficiently high standard so as to make it roadworthy again.
Category N: This signifies non-structural damage. The vehicle can be put back on the road if it is repaired to a roadworthy standard once again.

If you decide you would like to keep the vehicle and repair it yourself, the insurance company will normally payout the vehicle's value immediately prior to the accident, less of course any excess that you have to pay for yourself. You can then negotiate with the insurer to buy the car back from them. You cannot buy back a Cat A or Cat B since these will be deemed permanently unroadworthy and must be scrapped. In the case of a Cat C or Cat S you need to surrender your logbook (Vehicle Registration Certificate: or VC5) to the insurance company, which will forward it to the DVLA. Whilst you can still drive the car legally, provided that it is roadworthy, insured, taxed and MOT'd, you would need to apply for a new logbook.

Please note that the old Vehicle Identity Check (VIC), which made it necessary for owners of Cat C or Cat S cars to jump through (expensive) hoops to get a relacement log book, was abolished in 2015 so there is now no charge for this document.