Exactly as the title asks. Suppose I have a discrete distribution of which the only thing I know about it is the mean $\mu$ and variance $\sigma^2$. What would be the notation to describe this? Is there something that is commonly used that is similar to the normal distribution?
Some people write $X\sim F(\mu,\sigma^2)$ (for some unspecified distribution function, $F$) or occasionally $X\sim (\mu,\sigma^2)$ to indicate that $X$ is from some distribution with that mean and variance.
Or you can just say it in words, like "X has a distribution with mean $\mu$ and variance $\sigma^2$"
But to indicate it's discrete, you normally just say it's discrete. If you want to be specific about the values it can take, then you could take the trouble to specify them in mathematical notation. On occasion that can be tricky though.