Let $S$ be a smooth projective surface over $\mathbb{C}$ and $K_S$ be its canonical divisor (class). I know that $K_S^2$ refers to the self-intersection of the curve $K_S$ with itself.
What does it mean when $K_S^2=0$? This happens, for example, when $S$ is minimal and has Kodaira dimension $0$.
Also, by notation from Beauville's book page 93, what does it mean when $K_S=0$? Do we have $K_S^2 = 0 \implies K_S=0$?
$K_S^2=0$ tells you that $K_S$ and $-K_S$ are not big, i.e. $\kappa(S)\neq -\infty, 2$.
For your second question: if $K_S$ is trivial then $K^2_S=0$, but the other implication is false. Take a minimal Enriques surface $S$, then $K_S\neq \mathcal 0_S$, but it is torsion, namely $2K_S=\mathcal 0_S$. Therefore $4K_S^2=0$, i.e. $K_S^2=0$. On the other hand, for a minimal surface $S$ with $\kappa(S)=1$, $K_S$ is not torsion, but $K_S^2=0$.