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A project is a container for issues (the items that need attention for some reason). The project is mainly there to configure the issues grouped within it, so it defines their fields, screens, workflows, user permissions, notifications and so on. You can have different projects working in completely separate ways, all suited to the owners/users of the project.
A plan is an overview of a set of issues. It might be issues in a single project, or it may draw them in from many projects.
There is no project or plan hierarchy. But there is an issue-level hierarchy. Basic Jira has just Issue -> sub-task (although you can have many different types of issue or sub-task, there's only two layers). Jira Software implements Epic -> Issue -> sub-task. Advanced Roadmaps (which is probably what you are seeing here when you say "plan") gives us the ability to add more layers above Epic, most people being used to adding features, initiatives and themes.
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