The records of births, marriages and deaths for England and Wales are held by the General Register Office which is part of the UK Identity and Passport Service. The records themselves are only available in the form of certified copy certificates. The printed (and in some cases, hand written) indexes to the records have, historically, been available to the public but have now been withdrawn. Microfiche copies are avaiable in many locations and images of these indexes are also available through several web sites.
For further details see the Civil Registration sections of the England or Wales pages.
Records for Scotland are different and are at:
The General Register Office for Scotland
New Register House
For further details see the Civil Registration section of the Scotland page.
Irish records have their own system as well. The records of births, marriages and deaths for all of Ireland up to 1921, and for Eire since 1921 are held at:
8/11 Lombard Street East
Records for Northern Ireland since 1921 are at:
49/55 Chichester Street
Wales was largely conquered by the English in the 14th Century, and was officially united with England in 1535. King James VI of Scotland inherited the throne of England (as James I) in 1603, but the two countries remained separate entities until the Act of Union in 1707 forming 'Great Britain'. Ireland had been claimed by the English Kings since the 12th Century but was not officially joined to Great Britain until 1801. Ireland was given independence in 1921 as the Irish Free State (later called 'Eire'), but the counties of Ulster opted out and (as Northern Ireland) remain part of the United Kingdom.
So, today, the 'United Kingdom of Great Britain and Northern Ireland' includes 'Great Britain', which in turns includes England, Scotland and Wales.
The Channel Islands and the Isle of Man are 'Crown Dependencies' which are technically outside the United Kingdom and have their own local legislatures. This is course is the offical version. You will find many people referring to 'England' when they actually mean 'Britain' or the 'UK'.
Changes For This Version (1.1 - 1995/07/31)
Added location of Irish and Scottish records. Added explanation of the relationship between the various parts of the country.