JIRA is interesting for developers in two ways. 1) it is used to track tasks for software development (especially for agile projects) or as bug tracker (see JIRA User's Guide). 2) JIRA provides a large support for extending functionality with plugins written in Java (see Create a HelloWorld Plugin Project)
There's an installation guide for running a JIRA Server under Windows or Linux / Mac ( see Installing JIRA applications)
For local testing there's an SDK, which starts a JIRA test instance and can be used to develop JIRA Plugins (see Set up the Atlassian Plugin SDK and Build a Project)
The easiest way to run a local JIRA instance, is to install the SDK and run
For further details see Starting a local JIRA test instance
Please note that this runs a local test instance and is not meant for production.