Conditioning is a probabilistic operation that consists in examining the probabilistic properties of a random variable (or of an event) given the realised value of another random variable (or of an event)
Conditioning the random variable $X$ on the random variable $Y$ means deriving the conditional probability distribution of $X$ given $Y=y$, which is formally provided by the equation $$\mathbb{P}^X(X\in A)=\int \mathbb{P}(X\in A|Y=y)\text{d}\mathbb{P}^Y(y)$$ This covers as a special case the settings where $X$ or $Y$ (or both) are events indicators, e.g., $$ X=\mathbb{I}_A(Z)\quad\text{or}\quad Y=\mathbb{I}_B(W)\,. $$
When the pair $(X,Y)$ has a density $f_{X,Y}(x,y)$ against a reference measure and when $Y$ has a density $f_Y(y)$ against the projected measure, the density of the conditional distribution of $X$ given $Y=y$ is defined as $$ f_{X|Y}(x|y) = \dfrac{f_{X,Y}(x,y)}{f_Y(y)} $$ almost everywhere in $x$ and $y$.
