There are almost no laws that will stop the average chicken keeper from having a few birds in their back garden. That said there are a few things that are worth bearing in mind.
A chicken looking curiously at its owner
Protection of Animals Act and Animal Welfare Act
As with any animal, domestic or wild, cruelty to animals is illegal. With chickens you are required to provide adequate housing, food, water and care. Adequate housing is defined as housing that allows them to exhibit their normal behaviours (roosting, nesting, scratching and living in groups) and care can be understood as making sure your chickens health is well looked after.
If you do have over 50 birds then you need to register your flock with DEFRA - information on registering your poultry can be found on their website.
Under the 1950 Allotment Act you are allowed to keeping hens on your allotment, as long as they are only for the use of the tenants and not used for business or profit. However, if your chickens are regarded as a nuisance or a health hazard, or their welfare is affected they can be removed. No matter where your chickens are they are subject to welfare rights enforced by the Protection of Animals Act and the Animals Welfare Act. You can download the Welfare Of Animals On Allotments leaflet produced by the RSPCA on their site (note: when referring to hens, cockerels are excluded as they are more often covered by local law due to their noise).
By-Laws, Covenants and House Deeds
Occasionally there will be restrictions on a property or local area that bans the keeping of poultry. Check with your local council or property deeds if you suspect this may apply to you.
If you are living in rented accommodation it is worth checking your tenancy agreement to see if you are allowed to keep chickens. If you are unsure then contact your landlord directly and see what they say.