많이 – an adverb used with verbs (V a lot)
많은 – an adjective used with nouns (a lot of N)
우리가 외식을 할 때마다 친구가 음식을 많이 먹어요.
Whenever we eat out, my friend eats a lot of food.
많은 친구가 태국음식이 한국음식보다 더 선호해요.
A lot of my friends prefer Thai food to Korean food.
요즘 많은 사람이 상스러운 말을 해요.
These days many people use bad language.
그가 상스러운 말을 많이 해죠?
He uses bad language a lot, doesn’t he?
많다 is an adjective often found at the end of a clause:
한국에는 친구 많아요?
Do you have a lot of friends in Korea?
그는 걱정이 많아서 머리를 식히러 산책했어요.
He had a lot of anxieties so he went for a stroll to clear his head.
가보고 싶은 재미있는 곳이 많아요.
There are so many interesting places I want to visit.
(lit. interesting places that I want to visit are many)