If things are consecutive, they happen one after the other with no break. If there are five consecutive snowstorms on five consecutive days, you'll have to shovel your way out on day six.
Consecutive comes from the Latin consecutus, meaning "following closely" with no gap. Just like those snowstorms — one storm happened each day, back to back, for five days in a row. Consecutive numbers also follow each other, or advance in the right order. For example, 5, 6, 7, 8, 9, 10 are consecutive numbers. You should number the pages on a term paper to keep them consecutive so the teacher doesn't get confused.