Composer is an application-level package manager for the PHP programming language that provides a standard format for managing dependencies of PHP software and required libraries. It manages them on a per-project basis, installing them in a directory (e.g. vendor) inside your project.
An interface is a programming structure that allows the computer to enforce certain properties on an object. In object oriented programming, an interface generally defines the set of methods that an instance of a class that has that interface could respond to. It is actually a concept of abstraction and encapsulation.
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 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.