I guess that you created your workflow in a "Task" list that has the email alert to "Assigned To" enabled. You can disable this feature under Settings, List Settings, Advanced Settings.
Your approver likely has "Alert Me" features enabled for this list. You can delete these under Action, All Site Settings, User Alerts.
As a side note. Answer to the question this title sounds like it is asking: The "Send e-mail when ownership is assigned" feature in "Tasks" lists on SharePoint often causes problems by sending unwanted emails like the task has been changed alert this question refers to. For those of us who Own sites below the site collection level and do not have access to the top level. Here is a way to fix such problems.
Turn of the "Send e-mail when ownership is assigned" feature and design a workflow in SharePoint designer that mirrors this feature minus to portion of the functionality that is not wanted.