Any recent computer operating system (by recent, we mean since about 1970) has some sort of user interface—some way of specifying commands for the operating system to execute. But in lots of operating systems, that command interface was really built in and there was only one way to talk to the computer. Furthermore, an operating system's command interface would let you execute commands, but that was about all. After all, what else was there to do?
The Unix operating system popularized the notion of separating the shell (the part of the system that lets you type commands) from everything else: the input/output system, the scheduler, memory management, and all of the other things the operating system takes care of for you (and that most users don't want to care about). The shell was just one more program; it was a program whose job was executing other programs on behalf of users.
But that was the beginning of a revolution. The shell was just another program that ran on Unix, if you didn't like the standard one, you could create your own. So by the end of Unix's first decade, there were at least two competing shells: the Bourne Shell, sh (which was a descendant of the original Thomson shell), plus the C Shell, csh. By the end of Unix's second decade, there were a few more alternatives: the Korn shell, (ksh), and the first versions of the bash shell (bash). By the end of Unix's third decade, there were probably a dozen different shells.
You probably don't sit around saying "should I use csh or bash or ksh today?" You're probably happy with the standard shell that came with your Linux (or BSD or Mac OS X or Solaris or HP/UX) system. But disentangling the shell from the operating system itself made it much easier for software developers (such as Brian Fox, the creator of bash, and Chet Ramey, the current developer and maintainer of bash), to write better shells—you could create a new shell without modifying the operating system itself. It was much easier to get a new shell accepted, since you didn't have to talk some operating vendor into building the shell into their system; all you had to do was package the shell so that it could be installed just like any other program.
Still, that sounds like a lot of fuss for something that just takes commands and executes them. And you would be right—a shell that just let you type commands wouldn't be very interesting. However, two factors drove the evolution of the Unix shell: user convenience and programming. And the result is a modern shell that does much more than just accept commands.
Modern shells are very convenient. For example, they remember commands that you've typed, and let you re-use those commands. Modern shells also let you edit those commands, so they don't have to be the same each time. And modern shells let you define your own command abbreviations, shortcuts, and other features. For an experienced user, typing commands (e.g., with shorthand, shortcuts, command completion) is a lot more efficient and effective than dragging things around in a fancy windowed interface.
But beyond simple convenience, shells are programmable. There are many sequences of commands that you type again and again. Whenever you do anything a second time, you should ask "Can't I write a program to do this for me?" You can. A shell is also a programming language that's specially designed to work with your computer system's commands. So, if you want to generate a thousand MP3 files from WAV files, you write a shell program (or a shell script). If you want to compress all of your system's logfiles, you can write a shell script to do it. Whenever you find yourself doing a task repeatedly, you should try to automate it by writing a shell script. There are more powerful scripting languages, like Perl, Python, and Ruby, but the Unix shell (whatever flavor of shell you're using) is a great place to start. After all, you already know how to type commands; why make things more complex?
Why is this book about bash, and not some other shell? Because bash is everywhere. It may not be the newest, and it's arguably not the fanciest or the most powerful (though if not, it comes close), nor is it the only shell that's distributed as open source software, but it is ubiquitous.
The reason has to do with history. The first shells were fairly good programing tools, but not very convenient for users. The C shell added a lot of user conveniences (like the ability to repeat a command you just typed), but as a programming language it was quirky. The Korn shell, which came along next (in the early 80s), added a lot of user conveniences, and improved the programming language, and looked like it was on the path to widespread adoption. But ksh wasn't open source software at first; it was a proprietary software product, and was therefore difficult to ship with a free operating system like Linux. (The Korn shell's license was changed in 2000, and again in 2005.)
In the late 1980s, the Unix community decided standardization was a good thing, and the POSIX working groups (organized by the IEEE) were formed. POSIX standardized the Unix libraries and utilities, including the shell. The standard shell was primarily based on the 1988 version of the Korn Shell, with some C shell features and a bit of invention to fill in the gaps. bash was begun as part of the GNU project's effort to produce a complete POSIX system, which naturally needed a POSIX shell.
bash provided the programming features that shell programmers needed, plus the conveniences that command-line users liked. It was originally conceived as an alternative to the Korn shell, but as the free software movement became more important, and as Linux became more popular, bash quickly overshadowed ksh.
As a result, bash is the default user shell on every Linux distribution we know about (there are a few hundred Linux distros, so there are probably a few with some oddball default shell), as well as Mac OS X. It's also available for just about every other Unix operating system, including BSD Unix and Solaris. In the rare cases where bash doesn't ship with the operating system, it's easy to install. It's even available for Windows (via Cygwin). It's both a powerful programming language and a good user interface and you won't find yourself sacrificing keyboard shortcuts to get elaborate programming features.
You can't possibly go wrong by learning bash. The most common default shells are the old Bourne shell and bash, which is mostly Bourne shell compatible. One of these shells is certainly present on any modern, major Unix or Unix-like operating system. And as noted, if bash isn't present you can always install it. But there are other shells. In the spirit of free software, the authors and maintainers of all of these shells share ideas. If you read the bash change logs, you'll see many places where a feature was introduced or tweaked to match behavior on another shell. But most people won't care. They'll use whatever is already there and be happy with it. So if you are interested, by all means investigate other shells. There are many good alternatives and you may find one you like better—though it probably won't be as ubiquitous as bash.