A very practical question. I have seen this happening in most of the projects where I worked. There are multiple reasons for not completing the user story example-Dependencies, waiting for some inputs, defects FOUND, some unplanned work, team member went on unplanned leaves, not estimated properly etc.

As a Product Owner we can HANDLE this in below-mentioned ways:

• Split the user Story.

The User Stories which are not completed can be transferred to the next sprint.

The tasks which are completed for this user story can be closed. We can re-estimate the story again in the next sprint and take this story as a HIGH priority in the next sprint. 

• Product Owner may also say that unless all the tasks of a user story are completed we cannot accept. In this entire, a user story is transferred to the next sprint and re-estimated. (Not completed tasks.)

But the question arises Does the team earns the velocity credit?

If we are DEFERRING the unfinished user story to the next sprint then do not take the velocity credit for it. Let the user story meets its acceptance criteria and then take the complete credit in the next sprint. We already know that we use average velocity, this averages out and avoid the risk of overstating velocity.

But in case of splitting the user story then take the credit of points burned for the work completed and let the remaining points for the unfinished work in the next sprint. This just boosts up the team’s morale as they have been given the points for the work completed and efforts they have PUT in.

I am a bit reluctant in taking this decision. Take your call based on discussion.