Civil registration — the government recording of births, marriages and deaths — began in England and Wales in 1837 and in Scotland in 1855. Churches continued to maintain records after those dates, so if you don't find a civil record, check church records.
The Family Records Centre in London holds indexes to English and Welsh civil records, but you can order microfilmed indexes up to 1983 through an FHC. Do a place search for England or Wales in the FHL catalog, and look at the subject heading Civil Registration — Indexes. Since the indexes cover only three-month periods, locating a name can be hard if you don't have an approximate date. You'll have an easier time searching the following online indexes for English and Welsh records: