Cats and dogs do not require a permit to enter the U.S., they do not require a rabies vaccination either, as the UK and Ireland are rabies free.
You should however, get a health certificate from your veterinary stating that your pet is fit and healthy to travel. You should also contact your airline and the respective State Department of Agriculture (in the U.S.), to check if they require further documents from you and for guidance on how to register your pet once you have arrived.
For further information, please visit the Centers for Disease Control and Prevention.
Bringing pets to the UK is regulated by the UK Department for Environment, Food & Rural Affairs (DEFRA).