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 "provides 2558 links to websites that offer on-line transcriptions of UK births, marriages, deaths and censuses. A wide range of other indexes and transcriptions are also available for most counties, these may include parish records, wills, monumental inscriptions etc. "