When you do something unceasingly, you do it without stopping. Your dog might bark unceasingly every time you leave the house, keeping it up until you come home again.
The adverb unceasingly literally means "without ceasing," or "without stopping." To do something unceasingly also implies that you're pretty determined and focused as well. You might, for example, devote your life to working unceasingly to end child poverty and hunger. Both unceasingly and cease, or "stop," have a Latin root word, cessare, "go slow, be idle, leave off, or give over."