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.
Docker provides the ability to package and run an application in a loosely isolated environment called a container. The isolation and security allow you to run many containers simultaneously on a given host. Containers are lightweight and contain everything needed to run the application, so you do not need to rely on what is currently installed on the host. You can easily share containers while you work, and be sure that everyone you share with gets the same container that works in the same way.
By default Nginx is not capable of processing dynamic web pages with PHP so it requires the use of specific additional plugins for this purpose. One of these plugins is FPM (FastCGI Process Manager). FPM is an alternative PHP FastCGI implementation with some additional features (mostly) useful for heavy-loaded sites. It is the preferred method of processing PHP pages with Nginx and is faster than traditional CGI based methods.