Critical Value vs T-Distribution

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.

The $t$-distribution accounts for the extra uncertainty of estimating the population standard deviation with the sample standard deviation $s$. Its exact shape depends on the degrees of freedom: fewer degrees of freedom give fatter tails, and as they grow the curve approaches the standard normal. For example, a one-sample $t$ test with $n = 15$ uses the $t$ model with $15 - 1 = 14$ degrees of freedom. The test statistic is $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$, where $\bar{x}$ is the sample mean and $\mu_0$ the hypothesized mean.