# Question regarding conditional expectation [duplicate]

In Larry Wasseman's lecture notes(lecture 4, page 4) I found this statement

$$\mathbb{E}[Y|X=x] = \sum_y y f_{Y|X}(y|x)$$ or $$=\int_y y f_{Y|X}(y|x)dy.$$
An important point about the conditional expectation is that it is a function of $$X$$, unlike the expectation of a random variable (which is just a number). Usually, we use $$\mathbb{E}[Y|X]$$ to denote the random variable whose value is $$\mathbb{E}[Y|X=x]$$, when $$X=x$$. This is something that you should pause to digest.

Can someone please explain this statement a bit?
1. Specifically, in which cases one can make a mistake for misunderstanding this concept?
2. What is the difference in using $$X$$(RV) or $$x$$(its realization) in simplifying some expression in probability theory?