Modules are one of the simplest parts of Joomla and a great entry point for people learning to use the system (the equivalent of widgets in wordpress). They can be displayed just about anywhere on a page (in all the positions a template allows as well as in the main content area using the loadmodule plugin for the com_content component).
To understand how to install and use a module in Joomla it is recommended to read Administration of a Module in Joomla
Creating a simple module for Joomla is one of the simplest development steps you can do - and the creating a simple module tutorial is designed to take you through this. It starts with a simple module and then shows a few things you can then do with the module.
Introduced into Joomla 3.2 is a hidden component that allows modules to create AJAX requests. You can find documentation about using the AJAX component to help you create even better modules.