Installation is very easy, just use curl to download it and pipe to the php executable. I prefer to keep Composer inside my project directory but it is also possible to install globally.
That will create a file called composer.phar, which is the entire Composer executable script. At this point you have a couple of options. You can create a composer.json file manually or use './composer.phar init' to generate the file for you by answering a number of questions. You can then use then './composer.phar require
In my case the composer.json file looks something like this...
What that means is the name of my project is 'ikromin/travelblog.ws' and I am the author. In addition the project requires the jackalope/doctrine-dbal and the smarty libraries (specifying the version numbers).
Running './composer.phar install' will download and install all of the required libraries, including their dependencies, into your project.
Once that is done, it's just a matter of including the auto-loader like this...
...and you can begin to use the libraries. It's as simple as that!