You will need the Full Write Workflow permission or be an admin on the workflow containing the process you want to delete.
Navigate to the workflow containing the process you want to remove and click on the "X" button on the process you want to remove in the listing.
You can also delete from within the process by clicking on "Delete" from the "Actions" menu in the upper right.
A confirmation message will appear. Click on "Yes" to finish.
Helpful Tips
You cannot delete a process that is referenced by another process or task. For example, a process that has an "Add To Process" task that points to the process being deleted. You will need to first make sure no other references exist.