When things happen consecutively, they follow one after another with no gap and no jump. Days, years, games, page numbers: Monday, Tuesday, Wednesday are consecutive days; if it rains Monday through Friday without a break, it rained for five days consecutively. Numbers that run 3, 4, 5 are consecutive; 3, 5, 8 are not. Sports writing loves it (“won three titles consecutively”). It is a neat, slightly careful word. In speech people often just say in a row or one after another. It does not mean “so” or “because of that” — that is consequently, and mixing those two is one of the most common slips with this family.