Dogs, cats, ferrets, and horses can be brought into the UK without being quarantined as long as it is microchipped, has a "pet passport" or official veterinary certificate, and has been vaccinated against rabies. If you're traveling from an unlisted country (https://www.gov.uk/take-pet-abroad/listed-and-unlisted-countries) it will also need a blood test. Pet rabbits and rodents must spend four months in quarantine.
Note that several breeds of dogs are not permitted in the UK. All Pit Bull-types, Japanese Tosa, Dogo Argentino, and Fila Brasiliero dogs are illegal to own or bring in to the country.
Helpful Pet Resources
Pet Restrictions: What to know before you relocate your pet overseas.
?What you need to know on relocating your pet abroad.