번 is means 'times,' and 째 means 'after so many.' In other words, 번째 would be the article that makes the number into an "N-th." Meaning "after so many." So for ex; 17th, 18th, 19th, and so on. Hope this was useful! ^^

Just a modifier to make an ordinal number

둘 (Two) — 두 번째 (second) ... And so on

For "one — first" there is actually an exception:

하나 (One) — 첫 번째 (first)

번째 means "th". For example, 사(넷) means four and 네번째 means fourth.