First, let's look at statutes. The output of Congress is a stream of laws. The first law the comes out of Congress #X and is signed by the President is Public Law #X-1. The second one is P.L. #X-2, and so on. These public laws are collected and published in a series of books called the "Statutes at Large".
The Statutes at Large is not a convenient tool for legal research, since it is just a sequential listing of the laws passed by Congress and signed by the President. There's no organization by subject, so in theory you would have to look at everything starting at page 1 of the first volume up to the last page of the present volume (and then look at laws that have been passed since the last volume was printed...) and note which are relevant to the problem you are researching.
To make it easier to find law, private publishers took the Public Laws and organized them into a code. A code is basically a statement of the law, organized by subject rather than chronologically. These privately published codes had no official status. The official statement of the law remained the Statutes at Large.
In 1874 Congress made an official code of the US laws, and they updated it in 1878. These were authoritative, by which I mean if the code said one thing and the underlying Public Law said something else, the code version won. This meant that if you were researching a topic in, say, 1890, you would start at the 1878 code, and then only had to look at the Statutes at Large from between then and 1890, instead of having to go back to the very beginning.
Congress got its act together in the '20s, and started producing an official United States Code, and updating it every six years, with annual supplements. It's important to note that the USC is not automatically the official authoritative statement of the law. The law codified in the USC only supersedes the Statutes at Large when Congress explicitly says so. Congress does so on a title by title basis, by passing a law that basically says that title X of the USC is now the complete statement of a particular area of law, superseding all prior Public Laws in that area.
In addition to the official USC, published by the government, there are unofficial versions. The most important is the United States Code Annotated, published by a private company, West Publishing. It consists of the text of the USC code with, as you've probably guessed from the name, annotations supplied by West. The annotations give for each thing in the USC a list of appellate cases that have cited or construed that part of the court, along with a summary of what that case said. They also give legislative history information. The USCA is immensely useful. Suppose you have a question about fair use in copyright. You can read what the statute says in the USC or in USCA. But in USCA you will also see summaries of hundreds of court cases that have interpreted that statute--organized by West in a logical system based on what aspect of fair use they were construing.
Case law is similar. Courts issues opinions, and those opinions are officially collected and published in chronological order in a series of volumes. As with the similar Statutes at Large, this is not the best form for research.
Private publishers stepped in to address this. West Publishing is again one of the top players. They produced an outline of the law...a giant tree that breaks the law down into multiple levels of categories and sub-categories and assigns an identifier to each leaf. As court decisions come out, West takes them, identifies which areas of the law they touch on, and writes a short summary of what the court said about that area of law. This is published in a series of volumes, which is indexed by the leaf identifiers. There are other index volumes published that index this collection by time. The net result is that if you are interested in some particular area of law, you can find it in West's outline, get the relevant identifiers, and then go to West's index books and get pointed to the relevant cases. You can then look those up in West's books and read the notes to find out which cases you need to look at in detail.
That same outline and identifier system in West's case reporters is also used in West's USCA, so it all fits together.
Once you have found a relevant case, you need to find out of it is still good law. You don't want to build your argument around an appellate court decision that was later overturned by the Supreme Court. That is very embarrassing for a lawyer.
For this, you turn to a series of books from the Frank Shepard Company. These books list cases, and then tell you what other cases cite them and how they cite them (e.g., agreed with them, overturned them, distinguished them, and so on).
It sounds a bit awkward, but it actually works very well. With a good law library, you can reasonably answer any legal question concerning the law in your jurisdiction without too much flailing about. There are frequent supplements to the books (each book had a slip in the cover to insert a supplement), and cumulative indexes every so often so that you do not have to go through all the index volumes sequentially to find things.
If you need to research something outside your jurisdiction, it can be harder. A small county in Idaho, for instance, might not have Florida state court decisions in its library, so if for some reason Florida case law is relevant the lawyers or their researchers might need to go to a bigger library.