Having used a variety of issue tracker applications in the past, JIRA was how I always hoped other products worked. What comes "out of the box" is already easy to use and works well in most company workflows. Best parts of JIRA:
* Intuitive UI that offers a variety of ways to show issue and project status
* Multiple levels of taxonomy and organization for each issue type
* Customizable workflows
* Web-based app
* Integrations with other Atlassian products
* Markdown compatible
* Ability to tag others in comments
The major issue I have with JIRA is a common one across this whole eco-system, which is the inability to share tickets across more than one assignee. While this is a contentious workflow, since some people feel that there is no way for people to work on the EXACT same issue, I personally see a value in being able to assign a ticket to multiple people. At the very least, "epic" style issues should be able to be assigned to multiple people.
For JIRA specific issues, I am not satisified with their small-screen design. The majority of the functionality is not there, and the overall view is buggy and hard to use.
Using JIRA I have been able to not only keep track of all the things that I, and members of my team, are working on, but it is a great tool to organize the thought process when tackling large projects. Our agile workflow is easily parseable by using the built in agile view, which helps keep our burndown on target and our velocity on track.
Overall, using JIRA helps us to be able to know who is working on what, how much effort is required, and what is top priority across teams. We have noticed an increase in productivity and cross-team knowledge transfer from integrating JIRA in all our workflows.