I want to delete a project (or case/matter) but I don't see the "Delete" button in the Project Details.

Even though you may have User Permissions to create/edit/close projects, only a System Administrator can delete a project. 

Contact your System Administrator and ask them to delete the project for you. 

Also, please note you can not delete a project if there are any time, expenses, notes, and file attachments attached to the project. You would need to manually delete all entries from the Project in order to delete it. If you have time/expenses/invoices under the Project it would be best to just Close the Project instead of deleting it to keep the information on file.