At the moment there may not be a whole lot you can do by letting Cortana run in the background, however more features will come.
Currently Cortana periodically keeps checking your location to see if you've set any reminders for a specific location, it also updates th live tile every few minutes to alert you of breaking news or any odd weather patterns.
In the future there may be an ability for you to turn Cortana by saying a keyword even when the phone is locked, the keyword could be something like 'Hey, Cortana', but no one knows for sure.
It is worth noting that some people are experiencing battery issues by letting Cortana run in the background(and with the live tile.)
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