A Kit car is a car that has been built, usually by the owner, from a "kit" provided by the manufacturer, and from a Donor car, or from parts of several production cars. Ford Cortina's, Escorts, and Sierras are very common as donor cars because they are very available, and once they get older, are very cheap to buy.
Once all the parts are collected together the car can be built, then stripped down again and painted, before being rebuilt. Once the car is ready for the road it must be inspected. There are about a dozen SVA centers in the UK. The SVA (Single Vehicle Approval) test is like a super MOT. The test lasts about 3 hours and checks that the vehicle complies with all current regulations. This goes into such detail as the radius of curves on edges. Once the SVA has been passed the vehicle can be taxed and go on the road like any other car.
There is an a way of avoiding the SVA test. Some Kits retain the original chassis from the donor vehicle. These kits are classed as embodied cars and only have to undergo an MOT and an inspection by the local Vehicle Licensing Office, to check that the chassis number and engine numbers match those of the original vehicle.
The other way to avoid all these problems is to by a "kit" already built and on road. This is an acceptable option for the less mechanically minded, and those who do not have the time to build a kit. It can also be a cheaper option.
Insuring a Kit car is very easy. Whilst most companies are not happy with anything other that a standard "euro box" there are specialist companies who are not only happy to insure these cars, but often do so at a much lower cost than if the car was standard. In one well publicized case a Toyota MR2 was converted using a body kit, to look like a Ferrari 346, and in doing so the insurance DROPPED by £500. The reason for this is that the companies feel that if someone has spent hours of their own time building a car, they are less likely to do something stupid with it.
I have listed some companies in the links page.