Git is a version control system (VCS) software that is used for source code management (SCM). There are several examples of VCS software, such as CVS and SVN. What makes Git different is that it is a distributed VCS, that is, a DVCS.
Being a DVCS, when you use Git you create fully capable local repositories that can be used for offline work. When you get the files of a repository, you actually grab the full information (this makes the initial creation of local repositories out of a remote repository slower, and the repositories are bigger).
You can install git by installing the git package. You can test it by opening a terminal window, and running
git clone git://github.com/schacon/whygitisbetter.git
The files appear in a directory called whygitisbetter. In a subdirectory called .git/,git stores all the controlling information it requires to manage the local repository. When you enter the repository directory (whygitisbetter in our case), you can issue commands that will figure out what’s going on because of the info in .git/.
With git, we create local copies of repositories by cloning. If you have used CVS or SVN, this is somewhat equivalent to the checkout command. By cloning, you create a full local repository. When you checkout with CVS or SVN, you get the latest snapshot only of the source code.
What you downloaded above is the source code for the http://www.whygitisbetterthanx.com/ website. It describes the relative advantages of git compared to other VCS and DVCS systems.
Among the different sources of documentation for git, I think one of the easiest to read is the Git Community Book. It is consise and easy to follow, and it comes with video casting (videos that show different tasks, with audio guidance).
You can create local repositories on your system. If you want to have a remote repository, you can create an account at GitHub, an attractive start-up that offers 100MB free space for your git repository. Therefore, you can host your pet project on github quite easily.
GitHub combines source code management with social networking, no matter how strange that may look like. It comes with tools that allows to maintain your own copies of repositories (for example, from other github users), and helps with the communication. For example, if I create my own copy of the whygitisbetter repository and add something nice to the book, I can send a pull request (with the click of a button) to the maintainer to grab my changes!
If you have already used another SCM tool (non-distributed), it takes some time to get used to the new way of git. It is a good skill to have, and the effort should pay off quickly. There is a SVN to Git crash course available.
If you have never used an SCM, it is cool to go for git. There is nothing to unlearn, and you will get a new skill.
Git is used for the developement of the Linux kernel, the Perl language, Ruby On Rails, and others.