For discrete variables, $\text{E}(X|Y=y) = \sum x \ \text{P}(X=x|Y=y)$, while for continuous variables, $\text{E}(X|Y=y)= \int x \ f_X(x|Y=y) \ dx$, where the sum and integral are over the possible values taken by $x$.
An expectation conditioned on some subset of the possible values taken by $Y$, such as $\text{E}(X\mid a < Y < b )$, say, is also a conditional expectation.