How do I handle a backlog item that is only partially complete at the end of a sprint?

You can split the backlog item into two backlog items, one that stays assigned to the sprint and is completed in the sprint, and one that is unassigned from the sprint and contains the remaining work, to be assigned to a future sprint. To do this, select one or more partially complete backlog items, and select "Split these items" from the right-click menu. If a backlog item has tasks, the tasks that have not yet been started will be moved to the new backlog item, as part of the split. 

Here's a backlog item with tasks after it has been split:

You can also split backlog items manually by copying backlog items and moving tasks between them using cut, copy & paste.