In ideal situation all the User Stories should be completed by the end of the Sprint. But in reality because of multiple reasons we have some unfinished User Stories at the end of Sprint. Some teams move those unfinished User Stories to next Sprint. But the problem of this method is that we are actually changing the Scope of current Sprint while executing the Sprint and it is against the principle of Sprint. Sprint recommends that once you start the Sprint, the Scope of the Sprint should never change.
The other effect of moving the unfinished User Stories to next Sprint is on Burn Down Chart. As you move unfinished User Stories to next Sprint, Burn Down Chart will dramatically come down to zero. This Burn Down Chart will give false impression that the Team has Burned Down all the User Stories of current Sprint.
Please refer the Burn Down Chart below. It gives an impression that all the User Stories for a given Sprint were Burned Down.
If you see the Burn Up Chart of the same Sprint then you will see that the Scope of the Sprint is changed almost every day and at the end of the Sprint the Scope has reduced steeply. Ideally the Scope line of Burn Up chart should be straight line.
It is expected that in a Retrospective session, Team will discuss the reasons behind Unfinished User Stories and take corrective actions to avoid it in future. But if a Team refers above Burn Down Chart then there won't be any unfinished work and Team will miss an opportunity to learn and improve for next Sprint.
Instead of moving unfinished User Story to next Sprint, Team should Split unfinished User Story. By Splitting the User Story, we will keep the original User Story into current Sprint and move the unfinished part of User Story to next Sprint.By this method your Burn Down will have still some ToDo tasks at the end of Sprint for unfinished User Stories. But this is okay. Team can use that data for the Retrospective session.
Rally software provides nice feature of Splitting User Story. You can refer a URL below for the video of Splitting User Story in Rally.