What is Composer?
"Composer is a tool for dependency management in PHP. It allows you to declare the dependent libraries your project needs and it will install them in your project for you."
http://getcomposer.org/doc/00-intro.md
What is a dependency manager?
With a dependency manager, you can more easily load third-party libraries that your project uses. The Symfony framework makes extensive use of this by loading in e.g. Doctrine, Twig, Swiftmailer, etc, but you can also think about jQuery, Shadowbox plugin, etc. Normally, you should download these libraries manually and place them in the project and if there are updates, go through those same steps.
By using a dependency manager, you can specify which library to use and which minimum version to use. Running the install / update option checks which libraries are missing or outdated.