Critical Value vs Z-Score

A critical value marks how far out on a distribution you go to capture a chosen probability. For a confidence interval it is the number of standard errors that brackets the middle C% of the sampling distribution. For example, a 95% confidence interval for a mean using the normal model uses $z^* = 1.96$, because 95% of the standard normal curve lies within 1.96 standard deviations of the center. With small samples and unknown population spread you instead read $t^*$ from the $t$-distribution using the degrees of freedom.

A z-score standardizes a value by measuring its distance from the mean in standard deviations. You compute it as $z = \frac{x - \mu}{\sigma}$, where $x$ is the value, $\mu$ (mu) is the mean, and $\sigma$ (sigma) is the standard deviation. For example, a score of 85 in a distribution with mean 70 and standard deviation 5 has $z = (85 - 70)/5 = 3$, so it sits 3 standard deviations above the mean. A negative z-score means the value is below the mean.