If you buy a new car, the car dealer will normally register the car for you.
If you buy a used car, you need to fill out and sign the blue part of the Registration Document V5 to V5C before the seller sends it off to the Driver and Vehicle Licensing Agency (DVLA) to notify them of the change of keeper (i.e. the person who keeps the vehicle on a public road, not necessarily the legal owner). You should keep the green section of the document until you receive a full registration document from the DVLA.
If the registration document is unavailable, you can make an application for a registration document at the Post Office or get the 'application for registration certificate' online at www.dvla.gov.uk or www.direct.gov.uk. The DVLA will then send you a new registration document in your name.
You can drive a car that is registered in another country (including in other EU/EEA countries) for six months in the UK. After this period, you need to register the car in the UK. For details on registering your car, see the DVLA website.
The Highway Code, available from most UK bookshops and online at www.highwaycode.gov.uk, contains full details of all UK driving regulations and is essential reading for anyone planning to drive in the UK.