Civil Registration of births, marriages and deaths (known as "Vital Records" in some countries) started on 1 July 1837 for England and Wales. For further information see Civil Registration in England and Wales.
Here is a map showing the 1851 Registration Districts of Wales.. The Map is taken from The Religious census of 1851 : A Calendar of the returns relating to Wales, Vol 11, North Wales. Ed. by Ieuan Gwynedd Jones, UWP,1981
The UK BMD web site offers links to over 400 web sites that provide on-line information for UK births, marriages and deaths and censuses.