When you are searching for cases, your most efficient search will be if you have the case citation. Regulations also have their own citations as well. They all tend to follow a similar pattern:
Volume Number - Source Abbreviation - Section or Page
517 U.S. 308 = 517th volume of the U.S Reporter, section 308
116 S.Ct 1307 = 116th volume of the Supreme Court Reporter, section 1307
57 F.R. 41731 = 57th volume of the Federal Register, page 41731
If you are not sure what citation you are looking at, you can try searching Cardiff University's Legal Abbreviations Index (It covers case reporters and legal periodicals from the British Isles, the Commonwealth, and the United States) or come to the GPML Reference Collection and look at The Bluebook: A Uniform System of Citation (Harvard University) call number K70 .U4.
Paralell citations are instances when the same case or statute has been referenced/reported in more than one source. For example, a landmark case like Roe V. Wade (where decision of lower court was affirmed but then part of it was reversed - which actually is not considered good law) can be found in:
410 U.S. 113 = volume 410 of the U.S. Reporter, section 113
93 S. Ct. 705 = volume 93 of the Supreme Court Reporter, section 705
35 L. Ed. 2d 147 = volume 35 of the United States Supreme Court Reports, Lawyer's Edition, 2nd Series, page 147
This means that you can look in any of these reporters and find the same information reported. In the electronic sphere, if you go to LexisNexis Academic or WestlawNext you could use any of those citations for your search and you would get the same search result.