Running Tasks with Task Scheduler is one of most common scenarios in Administrators life. By default you can run your Task as SYSTEM which is OK solution if you don't have to access any resources over network. However if you do and you want to run it as specially created domain account for that you need to setup ‘Logon as Batch Job' on Windows. This privilege is granted through the Local or Domain Security Policy. If you don't do that and you try to run the task anyway you will get Event ID 4625 in Security log.
And in Task Scheduler history