The modern system of law reporting began in 1865. The Incorporated Council of Law Reporting started to publish the "official" Law Reports, currently published in four series: Appeal Cases, Queen's Bench Division, Chancery Division and Family Division. Other important series include the Weekly Law Reports, and the All England Law Reports. Series of reports specialising in particular areas of the law are also published, for example, the Criminal Appeal Reports. Only a tiny proportion (1-2%) of court cases are published (or reported) as law reports. You can find law reports on Lexis Library and Westlaw UK or in the LAWS sequence in Western Bank Library. Increasingly, court judgements are also freely available on the BAILII website www.bailii.org.
Before 1865, cases were reported by private reporters, and were known by the reporter's name. Collectively, these are known as the Nominate Reports. They have been reprinted in two series, the English Reports 1220-1865, covering the period 1220 - 1865, and the Revised Reports, covering the period 1785 - 1866. Both series are available in the LAWS sequence in Western Bank Library, and the English Reports are on HeinOnline and Westlaw UK. Most of these reports are now of historical interest, but some continue to be significant as precedents.