My professor defined a coupling as the joint distribution of $(X, Y)$ such that $X \sim \mu$ and $Y \sim \nu$ for probability measures $\mu$ and $\nu$.
I was not quite sure whether $X \sim \mu$ means $X$ is a random variable that induces the measure $\mu$, or $X$ is $\mu$-measurable.
You are given probability measure $\mu$ and $\nu$.
A coupling of $\mu$ and $\nu$ is any probability space that has random variables $X$ and $Y$ such that $X$ has distribution $\mu$ (i.e., $X\sim \mu$) and $Y$ has distribution $\nu$ (i.e., $Y\sim \nu$).
Here's an example of coupling.
Lemma. Let $\varphi$ be the characteristic function of a real-valued random variable $X$. Then, there exists a random variable with $|\varphi|^2$ as its characteristic function.
Proof. Note that $$ |\varphi(t)|^2 = \varphi(t)\overline{\varphi(t)} = \mathbb{E}\left[e^{itX}\right]\mathbb{E}\left[e^{-itX}\right]. $$ Let $\mu$ denote the measure induced by $X$. Create a new probability space containing random variables $X_1$ and $X_2$ such that $X_1 \sim \mu$, $X_2 \sim \mu$, and $X_1$ and $X_2$ are independent (you should convince yourself that this is possible). This new probability space is a coupling of $\mu$ with itself! Then, $$ |\varphi(t)|^2 = \mathbb{E}\left[e^{itX_1}\right]\mathbb{E}\left[e^{-itX_2}\right] = \mathbb{E}\left[e^{it(X_1 - X_2)}\right]. $$ That is, $|\varphi|^2$ is the characteristic function associated with $X_1 - X_2$.