I want to delete a branch both locally and remotely.
$ git branch -d
Now you can do it with the GitHub Desktop application.
After launching the application
Switch to the branch you would like to delete
From the "Branch" menu, select, "Unpublish...", to have the branch deleted from the GitHub servers.
From the "Branch" menu, select, 'Delete "branch_name"...', to have the branch deleted off of your local machine (AKA the machine you are currently working on)