Judicial decisions are identified by the names of the parties involved in the dispute and a reference to the law report(s) where the decision is published, e.g.
Barber v Somerset County Council  1 WLR 1089;  2 All ER 385
All cases on BAILII are also assigned "neutral" citations. For example, Barber v Somerset County Council  UKHL 13.
You can search BAILII with a neutral citation or with a published case report citation. Simply type the reference in the Find Case Citation box on BAILII's home page.
You do not need to include brackets around the date, and you can type all letters in lower case, e.g. to find Barber v Somerset County Council simply type
2004 1 wlr 1089 OR 2004 2 all er 385 OR 2004 ukhl 13