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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.
The $|$ symbol in probability theory stands for “given”. You would most commonly see it used for conditional probability $P(Y|X)$, the probability of $X$ given $Y$. While it’s a slight abuse of notation, you could see something like
$$ Y|X \sim \mathcal{N}(\mu, \sigma) $$
for $Y$ conditionally on $X$ following normal distribution. You would also see it to show some properties of distributions, like conditional expectations $E[Y|X]$, or variance $\operatorname{Var}(Y|X)$, etc.
Notice that something like $X|Y$ alone doesn’t make much sense. What would it be? “A random variable conditional on another random variable”? Conditioning is about the perspective you take when looking at the variable, not a property of the variable. You can “transform” conditional probability to joint, or marginal, or reverse it (Bayes theorem) with just simple mathematical manipulations on the distributions.
This can get quite philosophical fast. But, Judea Pearl's book Causal Inference for Statistics Section 1.3.3 provides a nice intuition, the operator $|$ implies a filtering of the data in the frequentist interpretation. An intuitive example would be two variables having bounds $P(X>a|Y<b)$, so conditioning implies filtering the data, i.e., removing parts of the data where condition $Y<b$ doesn't hold first.
Regarding if $X|Y$ is an event or not. It isn't an event in plain form but the resulting filtering operation leads to an event. Philosophical part of this then, what would be the Bayesian interpretation and even conditional probability can exist in isolation.
