Enhanced probability.
(Definition of $Y|X$.) The random variable $Y|X$ (read: $Y$ given $X$) is a function of $x$ and $y$. Given an $x$, the value is $y$ with probability $P(Y=y | X=x)$.
$E[Y|X]$, sometimes written $E_y[Y|X]$ is a function of x, giving expected value of y when you know $X=x$. That is, $$E_y[Y|X] = \sum_y P(Y=y | X=x)\cdot y .$$
Law of Total Expectation. Using the definitions, try to prove that $$ E[Y] = E_x [ E_y [ Y|X] ]. $$ The left hand side could be written $E_y$ as well, if that is not confusing. Some books write this without any subscripts, which I find hard to read.