Composer is not a package manager in the same sense as Yum or Apt are. Yes, it deals with "packages" or libraries, but it manages them on a per-project basis, installing them in a directory (e.g. vendor) inside your project. By default it does not install anything globally. Thus, it is a dependency manager. It does however support a "global" project for convenience via the global command.
This idea is not new and Composer is strongly inspired by node's npm and ruby's bundler.
Suppose:
You have a project that depends on a number of libraries.
Some of those libraries depend on other libraries.
Composer:
Enables you to declare the libraries you depend on.
Finds out which versions of which packages can and need to be installed, and installs them (meaning it downloads them into your project).
You can update all your dependencies in one command.
See the Basic usage chapter for more details on declaring dependencies.