Things that occur one after the other, with no break between, happen consecutively. A baseball team that wins three games in a row has won three games consecutively.
You can use the adverb consecutively whenever you talk about things or events that follow each other without interruption. If your Chihuahua wins Best in Show at the dog show for three years consecutively, it means that she won it in Year 1, Year 2, and Year 3, without missing a year. The word can also describe things organized in numerical order: "Make sure you stack the pages of your manuscript consecutively before you submit it to the magazine." The Latin root of consecutive means "to follow."