When you carry something over a longer period of time than originally planned, that's a continuation. The City Council voted on a continuation of their discussion on the school budget after failing to reach an agreement today.
You can also think of a continuation as the act of continuing to do something or to be in a particular situation. The continuation of your allergies makes you suspect they might have something to do with your new cat. Even after your cousin loses his job, he's grateful for the continuation of his health insurance. The Latin root, continuationem, means "a following of one thing after another."