All users except collaborators can delete projects by sending them to the Recycle bin. Only account admins can permanently delete projects.
When you delete a project that has tasks, subtasks, subfolders, and subprojects:
If those items are only located in that project then they're deleted when the parent project is deleted.
If those items are stored in additional locations, then they'll continue to live in those additional locations even after the project is deleted.
Deleting a project moves it to your Recycle bin where it can be permanently erased or restored. This can be done in two ways.
Navigate to the space containing the relevant project.
Right-click a project from the left-hand navigation panel.
Select Delete from the drop-down list that appears.
Click Yes in the confirmation window that appears.
Alternatively you can:
Yes, admin users can permanently delete projects. After navigating to the Recycle bin they can right-click a project title in the left-hand navigation panel and click Erase. Alternatively they can open the project info panel from the Recycle bin and click Erase in the top right corner of the panel.