There is an important reason for having the citation number. There are many instances in which there is more than one case with the exact same name. For example, if you wanted to look up the Supreme Court case of Miller v. California, you would find there are 4 cases (one [389 U.S. 968] decided in 1967, one [392 U.S. 616] in 1968, one [413 U.S. 15] in 1973 and one [418 U.S. 915] in 1974), involving different people named Miller, and all involving different issues. By having the citation number, you can look up exactly which case you were interested in.
These examples are all from the U.S. Supreme Court. Let's examine Miranda v. Arizona 384 U.S. 436 (1966) and show what all the elements are. Some of this may be obvious but I'll explain all of it.
In the case of Griswold v. Connecticut, there are two other references, "S.Ct." which refers to a private company's reprinting of Supreme Court Decisions which is called the "Supreme Court Reporter". "L.ed.2d" refers to the second edition of the "Lawyers Edition," another privately printed set of volumes of court decisions. That case can be found in Volume 381, Page 479 of the Official Supreme Court records, in Volume 85, page 1678 of the private Supreme Court Reporter, and also Series 2, Volume 14, Page 510 of the Lawyers Edition.
So, as another example, for Roe v. Wade, 410 U.S. 113 (1973) if we went to a law library, we would find the case (which was decided in 1973) in volume 410 of the decisions of the U.S. Supreme Court, starting on page 113.
See also: Landmark case