A case is usually referred to using a case citation.
A law report citation tells you where to find a case in a law report series:
In January 2001, the official neutral citation system was introduced. It's 'neutral' because the citation is independent of any law report series and instead refers directly to the court:
You can find out more about neutral citations on the BAILII website.
A law report series will usually be referred to using an abbreviation, e.g. WLR for Weekly Law Reports.
To find out what an abbreviation means, check the Cardiff Index to Legal Abbreviations or Raistrick's Index to Legal Citations and Abbreviations.
To find out what happened in a case, you can consult a law report or a court transcript. Over time, reports of cases have grown into a body of case law.
Law reports are made available by publishers. They pay law reporters to sit in court and transcribe what is said when a case is heard. However, not all cases get reported (= published). In fact, only 1-2% of cases heard in UK courts each year get reported. To be reported, a case must raise a point of legal significance.
Also, the majority of reported cases are from the higher courts (Supreme Court, Court of Appeal, High Court). You can find out more about the UK court structure on the Courts and Tribunals Judiciary website.
More recently, official court transcripts of cases have been made freely available on the Internet.
You can find many law report series in print on the second floor of the Sir Michael Cobham Library, including:
How do I find a case in a printed law report series? To find: R v Green  1 AC 1053 (HL), first I check what 'AC' means (see 'What is a law report abbreviation?' in the left column). It's an abbreviation for The Law Reports, Appeal Cases. I find this law report series on the library shelves. I locate the volumes for 2008, then I take volume 1 off the shelf and turn to page 1053.