What does $X=x|Y=y$ actually mean by itself? I know that:
$P(X=x|Y=y)=\frac{P(X=x,Y=y)}{P(Y=y)}$
But I'd like to understand what $X=x|Y=y$ means by itself, for example $(X=x,Y=y)$ means $X=x$ and $Y=y$. Would $X=x|Y=y$ mean? that $X=x$ when $Y=y$?
A example to make it easier (take these as observations):
$X=\{1,6,7,8\}$
$Y=\{0,1,19,5\}$
$(X=x|Y=y) = ?$

Example: Say you have a group of men and women and know their handedness (left/right). It is like depicted in the table below
$$\begin{array}{r|c|c | c}
&\text{men}&\text{women} &\text{total}\\
\hline
\text{left handed}&9&4&13\\\hline
\text{right handed}&43&44&87\\\hline
\text{total}&52&48&100
\end{array}$$
Say you pick randomly a person out of this group then it is $13\%$ probability that they are left handed. But if you know that the person is a woman, then the probability is $4/48 \approx 9 \%$.
To express this latter case, the probability of an event, given another event or condition, one uses the vertical bar symbol $\vert$.
$$P(X\vert Y) = \text{probability of event $X$ given/conditional on event $Y$}$$
So it is about both events $X$ and $Y$ happening. But, this is different from $P(X,Y)$, the probability that both $X$ and $Y$ are happening.
The probability for left handedness given that a person is a woman, is not equal to $4 \%$ the probability that someone is a woman and left handed.

The expression $X\vert Y$ occurs within the probability operator $P()$. But you should not read all the contents as a single event.

*

*So this is not how you must interpret it: "$P(\dots)$ is probability of the event on the dots. So $P(X\vert Y)$ is the probability of the event $X\vert Y$." This $X\vert Y$ is not an event (as Henry noted in the comments). The vertical bar $\vert$ adds additional parameters to the probability operator and refers to conditions.

