One easy way to understand "ABI" is to compare it to "API".
You are already familiar with the concept of an API. If you want to use the features of, say, some library or your OS, you will program against an API. The API consists of data types/structures, constants, functions, etc that you can use in your code to access the functionality of that external component.
An ABI is very similar. Think of it as the compiled version of an API (or as an API on the machine-language level). When you write source code, you access the library through an API. Once the code is compiled, your application accesses the binary data in the library through the ABI. The ABI defines the structures and methods that your compiled application will use to access the external library (just like the API did), only on a lower level. Your API defines the order in which you pass arguments to a function. Your ABI defines the mechanics of how these arguments are passed (registers, stack, etc.). Your API defines which functions are part of your library. Your ABI defines how your code is stored inside the library file, so that any program using your library can locate the desired function and execute it.
ABIs are important when it comes to applications that use external libraries. Libraries are full of code and other resources, but your program has to know how to locate what it needs inside the library file. Your ABI defines how the contents of a library are stored inside the file, and your program uses the ABI to search through the file and find what it needs. If everything in your system conforms to the same ABI, then any program is able to work with any library file, no matter who created them. Linux and Windows use different ABIs, so a Windows program won't know how to access a library compiled for Linux.
Sometimes, ABI changes are unavoidable. When this happens, any programs that use that library will not work unless they are re-compiled to use the new version of the library. If the ABI changes but the API does not, then the old and new library versions are sometimes called "source compatible". This implies that while a program compiled for one library version will not work with the other, source code written for one will work for the other if re-compiled.
For this reason, developers tend to try to keep their ABI stable (to minimize disruption). Keeping an ABI stable means not changing function interfaces (return type and number, types, and order of arguments), definitions of data types or data structures, defined constants, etc. New functions and data types can be added, but existing ones must stay the same. If, for instance, your library uses 32-bit integers to indicate the offset of a function and you switch to 64-bit integers, then already-compiled code that uses that library will not be accessing that field (or any following it) correctly. Accessing data structure members gets converted into memory addresses and offsets during compilation and if the data structure changes, then these offsets will not point to what the code is expecting them to point to and the results are unpredictable at best.
An ABI isn't necessarily something you will explicitly provide unless you are doing very low-level systems design work. It isn't language-specific either, since (for example) a C application and a Pascal application can use the same ABI after they are compiled.
Edit: Regarding your question about the chapters regarding the ELF file format in the SysV ABI docs: The reason this information is included is because the ELF format defines the interface between operating system and application. When you tell the OS to run a program, it expects the program to be formatted in a certain way and (for example) expects the first section of the binary to be an ELF header containing certain information at specific memory offsets. This is how the application communicates important information about itself to the operating system. If you build a program in a non-ELF binary format (such as a.out or PE), then an OS that expects ELF-formatted applications will not be able to interpret the binary file or run the application. This is one big reason why Windows apps cannot be run directly on a Linux machine (or vice versa) without being either re-compiled or run inside some type of emulation layer that can translate from one binary format to another.
IIRC, Windows currently uses the Portable Executable (or, PE) format. There are links in the "external links" section of that Wikipedia page with more information about the PE format.
Also, regarding your note about C++ name mangling: When locating a function in a library file, the function is typically looked up by name. C++ allows you to overload function names, so name alone is not sufficient to identify a function. C++ compilers have their own ways of dealing with this internally, called name mangling. An ABI can define a standard way of encoding the name of a function so that programs built with a different language or compiler can locate what they need. When you use extern "c" in a C++ program, you're instructing the compiler to use a standardized way of recording names that's understandable by other software.
