Composer is a tool for dependency management in PHP. It allows you to declare the libraries your project depends on and it will manage (install/update) them for you.
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.
Composer requires PHP 5.3.2+ to run. A few sensitive php settings and compile flags are also required, but when using the installer you will be warned about any incompatibilities.
To install packages from sources instead of simple zip archives, you will need git, svn, fossil or hg depending on how the package is version-controlled.