Structure plugin adds to JIRA the ability to organize issues in a hierarchy (of any depth).
In short, its main feature is the solution of a single very important problem in JIRA: Subtasks cannot have subtasks of their own. While certain hacks exist, they don't fully solve the problem. Structure plugin does.
The issue structure, provided by the plugin, brings JIRA closer to project management tools like Microsoft Project. You can see the big picture and dig into details as needed.
At the same time, breaking down issues into pieces can be helpful at the micro level, letting you streamline your thoughts and control the task progress.
If used at the extreme, issue hierarchy can replace project versions, or components, or other categorization tools.
Structure plugin lets you use JIRA as a mindmapping tool, or as a todo list.
The very neat and handy Structure widget speeds up working with a list of issues even if you don't need the structure.
And collaboration features let your team work on the same list of issues remotely.