いっぱい：noun / adverb : *(full of )? たくさん: noun / adverb : *(a pile of )? 多い：adjective : *(many, much )?

いっぱい：I use this word as somewhat colloquial. I do not use this word for my boss. Sort of chatty word depending on a situation. たくさん：You can use this word in any situation. おおい：It means the same with "たくさん". It become more formal depending on how you use "おおい".

For your another question below. We usually use "の" as "of", 多くの人々(a lot of people), for instance. So noun is used after の On the other hand, 多い tends to be put in the end such as 人が多い. 多い describes the noun before it, and noun rarely be used after it. However, 人が多い can be used as adj. Let's say 人が多い町(town), but never be 多い町. This time you need -が.