You are correct that
\begin{equation*}
p(y|\mathbf{x})=\dfrac{p(\mathbf{x},y)}{p(\mathbf{x})}.
\end{equation*}
Similarly, we can write the joint probability $p(\mathbf{x},y)$ as follows:
\begin{equation*}
p(\mathbf{x},y)=p(\mathbf{x}|y)\cdot p(y)
\end{equation*}
From the above two equations, we obtain
\begin{equation*}
p(y|\mathbf{x})=\dfrac{p(\mathbf{x}|y)\cdot p(y)}{p(\mathbf{x})}
\end{equation*}
In the context of supervised learning, the variable $y$ is used to denote the class labels, and the vector $\mathbf{x}$ for measurement vector or feature vector. For the purpose of discussion, let us assume that the class label $y$ takes values in the set $\{1,2\}$, where $1$ denotes male and $2$ denotes female. Similarly, $\mathbf{x}$ is a measurement vector on two variables say, $(x_{1},x_{2})$, where $x_{1}$ stands for height and $x_{2}$ stands for weight of individuals.
$p(y|\mathbf{x})$ denote the posterior density for $y$ given the observation $\mathbf{x}$. For example, $p(1|\mathbf{x})$ means that given the observation $\mathbf{x}$, what is the probability that sample belongs to the class males. Similarly, we can interpret $p(2|\mathbf{x})$.
$p(\mathbf{x}|y)$ stands for the class conditional probability density. For example, $p(\mathbf{x}|1)$ denote the probability density for males and $p(\mathbf{x}|2)$ denote the probability density for females respectively. In supervised learning, these class conditional densities are usually known in advance. Finally $p(y)$ denote the prior probability for the class label $y$. For example, $p(1)$ denotes the probability that an individual/example selected randomly from the population is a male.
Assuming that, the prior probabilities for the classes and class conditional densities are known, Bayes rule says, assign an observation $x_{0}$ whose class label is not known, to the class, say, male, if
\begin{equation*}
p(\mathbf{x_{0}}|1)p(1)>p(\mathbf{x_{0}}|2)p(2).
\end{equation*}
Note that, $p(\mathbf{x})$ is not used in the classifier. Classification rules usually require class conditional densities.
