I was asked if the affine variety corresponding to $\mathbb{C}[X,Y]/(XY)$ is irreducible. I am pretty sure that it is not, because it would contain varieties given by $1+x+x^2+\cdots$ and $1+y+y^2+\cdots$, so then it could be further decomposed. Is that correct?
However, if we have $(XY-1)$ instead of $(XY)$, then it would be irreducible, since $xy-1$ is an irreducible polynomial?
You are correct. Your first variety not irreducible precisely because $xy$ is not irreducible in $\mathbb{C}[x,y]$. The irreducible components of your variety are $\{x=0\}$ and $\{y=0\}$.
Your second variety is irreducible because $xy-1$ is an irreducible polynomial.