As with so many things, legal citations to case law is a confusing mish-mash of historical systems. (And, also like so many things, there are numerous oddities, inconsistencies, and exceptions, so this is just a general discussion).
Legal decisions were, and still are though they are less commonly used in the internet age, printed in series of bound volumes, some put out by private companies, some put out as official government publications.
For instance, the West Publishing Company (the biggie in case law publishing) originally published federal Court of Appeals decisions in the Federal Reporter. A decision would be referred to by the volume number then “Fed.” indicating the Federal Reporter, and then the page number the decision started on, for instance 220 Fed. 405. To find that decision, you would pull volume 220 of the Federal Reporter off the shelf, and open to page 405 to find the case. After the Federal Reporter reached an unwieldy number of volumes, West started issuing the Federal Reporter, Second Series, and later Third Series, abbreviated as F.2d and F.3d. For instance, the Jordan v. City of New London case you cite, 225 F.3d 645, can be found starting on page 645 of volume 225 of the Federal Reporter, 3rd Series.
Often the same cases will be published in multiple reporters, often an official government reporter and a private reporter. For instance, U.S. Supreme Court decisions are published in United States Reports (abbreviated U.S.), an official publication issued by the federal GPO, as well as West’s Supreme Court Reporter (abbreviated S.Ct.), which is published more quickly than the official government reporter, and also has editorial enhancements such as headnotes, which make legal research easier using that format.
States have a mix of similar reporting systems, with West having a National Reporter System of books that cover all decisions “officially” issued for publication, with many states having their own reporters (though some use the West reporters as their official case books). In most states, all decisions of the states highest appeals case, and most or all of the decisions of the intermediate appellate courts, are officially reported, and some decisions at the trial court level are. The West reporters are regional, with titles like “Northeast Reporter,” or “Southern Reporter,” though because of their volume of case law, California and New York have their own “Supplement” series of reporters, which include lower court decisions from those states, with the highest court decisions reported both in the supplement and the regional reporter.
Currently, most everything is online, but officially reported cases are still referenced by the volume and page numbers in the official books.
This brings to “unreported” cases, meaning cases not published in the official case books. Because there are many more decisions than can be reasonably published in bound volumes, particularly at the lower court level, many cases are published in electronic form only. There are two, large commercial case reporting databases, Westlaw (from West Publishing) and Lexis, as well as some smaller services. Westlaw takes all materials published on their service dating from a given year, and gives them a citation number starting with the year, then “WL” and then a serial number based on all of the documents added to the service for that year. For instance, 1999 WL 780977 is the 780977th document from 1999 added to Westlaw. Lexis lists documents by year and category, such as 1999 U.S. Dist. Lexis 2345, meaning it is the 2345th U.S. District Court decision from 1999 added to the database.
In addition, many states have online databases of their opinions. For instance, the New York Reporter of Decision (an official of the court system) indexes decisions and posts them online under a New York Slip Opinion number (N.Y. Slip Op.), including official decisions from the appellate courts (and some officially reported lower court decisions), as well as .pdf scans of many trial court opinions that have not been officially reported.
Finally, and this does not really involve the citation of legal decisions, each court will give each case or appeal before it a unique identifying number, often known as a Case Number or an Index Number. Federal District Courts typically give cases numbers in the form of “97-CV-1012-PCD,” where 97 means the case was filed in 1997, CV mean it is a civil case (as opposed to CR for criminal, and some other codes), 1012 means it is the 1012th civil case filed that year, and PCD are the initials of the assigned judge (the 3: is a code used in some districts for what courthouse or division the case is in). Federal appeals courts usually give appeal numbers like “99-9188,” which means that the case was filed in 1999 and given number 9188.
Services like Westlaw have a function that (sometimes imperfectly) tracks the history of decisions issued in a case, including appeals (sometimes multiple appeals), remands, and other decisions issued. And, typically, where an appeal reviews an officially-reported lower court decision, it will note the citation of that decision. Otherwise, if a reported appeal decision lists the case number of the lower court case, often you can go to the docket of that case in the lower court (particularly where cases are under a publicly available electronic filing system) and find the decision that is being reviewed.