Added to something that already exists, in order to make it more complete or better. The main thing is still there — the supplementary thing comes on top of it. "supplementary materials", "a supplementary payment", "supplementary information".

It is a formal word, at home in schools, offices, government forms and academic writing. In everyday talk people just say extra or additional. But there is a shade of difference worth hearing: extra means simply "more", while supplementary means "more, and it fills a gap in the main thing". A supplementary lesson is not just another lesson — it covers what the main lessons did not.

A few places you will meet it often: supplementary reading (books beyond the textbook), a supplementary question (a follow-up question after the first answer, common in parliament and interviews), supplementary benefits or supplementary income (money added to what you already earn), and supplementary exams in some school systems (a second chance exam).

It can also be a noun in British use, meaning a supplementary question, but a learner rarely needs that.

American writing often prefers supplemental with almost the same meaning; British writing prefers supplementary. Do not spend energy on the difference.