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I may have missed this during my intro stats/prob course but what is the difference between:

$E_Y[X]$ and $E[X|Y]$?

It seems like one you are marginalizating over and the other you are conditioning on. I keep seeing the following:

$E[X] = E_Y[E[X|Y]]$. I do not understand, intuitively, why this is true. Can someone please explain this to me?

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    I'm not sure the $E_Y$ in $E(X) = E_Y(E(X|Y))$ is anything other than a reminder that what's inside is a random variable depending on $Y$, which I could see being intuitively helpful. $E_Y(X)$ has no meaning to me. I would just write $E(X) = E(E(X|Y))$ which is a well defined thing since $E(X|Y)$ is a random variable on the same probability space as $X$ and $Y$. – 2017-01-13

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Here is some intuition. In the inner expectation, you have $E(X\mid Y)$, which is a function of $Y$. You are conditioning on $Y$, and then you integrate out $X$, leaving you an expression dependent only on $Y$. Then you take the expected value with respect to the distribution on $Y$ to recover the original expected value, which no longer depends on the random variables $X$ and $Y$.