What does "parameterized by" mean? Sometimes I have seen likelihood written as $L(\mu,\sigma |y)$ and sometimes as $L(y|\mu,\sigma)$.
I have been told that in the first case it means that there is a pre-assumed model depicting the probability density of $\mu$ and $\sigma$ parameterized by $y$. In the second case it means the pre-assumed model depicts the probability density of $y$, parameterized by $\mu$ and $\sigma$
I have seen the $|$ symbol used before in Bayes theory and read it as "given that"
I am from a programming background (C#) and tend to think of parameters as inputs to a function.
How should I be thinking about this word in statistics?

“Parametrized by” means that the function $f$ of $x$ has additional parameters $\Theta$, so we can write it as $f(x;\Theta)$. In such a case, we call the function by $x$ given a fixed value of $\Theta$.
Likelihood has two meanings, the traditional on, and Bayesian. Traditionally, likelihood is written as
$$
L(\theta|x) \propto P(x|\theta)
$$
The vertical bar $|$ is used on the right-hand side to denote conditional probability and is a slight abuse of notation on the left-hand side. People write it as $L(\theta|x)$ to show that we keep the data $x$ fixed, but we evaluate the function for different parameters $\theta$. In a Bayesian setting, you usually would not see the left-hand side notation, just the right-hand side will be called the likelihood (here you might have seen $L$ used instead of $P$). See Wikipedia entry on likelihood seems ambiguous for more discussion.
