"Category" is that central thing of the science of nodes and arrows between them.

"Functor" is a mapping - a function - from a category to another category, where each node is mapped and each arrow is mapped according to node mapping. Like, if 1 in number category maps to A in character category, 2 maps to B, and there is an arrow 1->2 in number category, then this arrow should map to arrow A->B in character category.

"Endofunctor" adds that the resulting category should in fact be the same as original. So, 1->2 can e.g. map to 3->4 in the same number category.

"Monoid" is requirement that with the category we also have defined binary operation (let's call it X), that the operation is associative and also that category has "identity" node (let's call it I) - that is, for any A both operations A X I and I X A both produce A. It's like adding with zero.

Now, this is not enough to understand monads - but it provides a foundation for what those monads, mathematically speaking, are. Consequences of these definitions can lead far :) .