To ensure the fraction $\frac{{x-2}}{{x-3}}$ is meaningful, the denominator must not be equal to zero. Thus, we set the denominator not equal to zero and solve for $x$:

1. Start with the condition for the denominator: $x-3 \neq 0$.
2. Solving this inequality gives us the condition for $x$: $x \neq 3$.

Therefore, the condition that $x$ should satisfy to make the fraction meaningful is $x \neq 3$.

So, the correct answer is $\boxed{C}$.