This notation has little to do with Bayesian statistics. The notation $$p(y|\mu,\sigma^2)$$ means that this density (I suppose this is a density, but it could be a cdf as well) is indexed by two parameters $\mu$ and $\sigma^2$, as for instance in the Normal density $\text{N}(\mu,\sigma^2))$. One could have used $$p(y;\mu,\sigma^2)\quad\text{or}\quad p_{\mu,\sigma^2}(y)$$instead.

When $\mu$ and $\sigma^2$ turn into random variables, as in Bayesian statistics, this becomes a conditional density of a random variable $Y$ with realisation $y$ given the random vector $(\mu,\sigma^2)$. If conditional density is new to you, you should first check an introductory probability book or the first chapters of Casella and Berger.

This notation has little to do with Bayesian statistics. The object $$p(y|\mu,\sigma^2)$$ is a density for the random variable $Y$ taking values $y$; the second part "|μ,σ²" means that this density (I suppose this is a probability density (pdf), but it could be a cdf as well) is indexed by two parameters $\mu$ and $\sigma^2$, as for instance in the Normal density $\text{N}(\mu,\sigma^2))$. Changing $(\mu,\sigma^2))$ modifies the density function. One could have used $$p(y;\mu,\sigma^2)\quad\text{or}\quad p_{\mu,\sigma^2}(y)$$instead. (Incidentally the bar "|" notation was introduced by Harold Jeffreys in the 30's, the same influential Bayesian Jeffreys as in Jeffreys' prior.)

When $\mu$ and $\sigma^2$ turn into random variables, as in Bayesian statistics, this becomes a conditional density of a random variable $Y$ with realisation $y$ given the random vector $(\mu,\sigma^2)$. If the concept of conditional density is new to you, you should first check an introductory probability book or just the first chapters of Casella and Berger for instance.