The Central Limit Theorem
If a random sample of size $$n$$ is taken from any distribution with mean $$\mu$$ and variance $$\sigma^2$$, the sampling distribution of the mean will be approximately $$\sim N(\mu,\sigma^2/n)$$, where '$$\sim$$' means 'is distributed as'. The larger $$n$$ is, the better the approximation.
The standard normal and Student's t distributions
If a random variable $$X \sim N\left(\mu,\sigma^2/n\right)$$ then $$~Z = (X-\mu)/\sigma \sim N(0,1)$$, the standard normal distribution.
The $$t$$ distribution with $$(n-1)$$ degrees of freedom is used in place of $$Z$$ for small samples size $$n$$ from normal populations when $$\sigma^2$$ is unknown. As $$n$$ increases the distribution of $$t$$ converges to $$N(0,1)$$. These distributions are used, for example, for inference about means, differences between means and in regression.
Fisher's $$F$$ distribution
If $$X_1 \sim \chi^2_{\nu_1}$$ and $$X_2 \sim \chi^2_{\nu_2}$$ are independent random variables then \[\frac{X_1/\nu_1}{X_2/\nu_2}\sim F_{\nu_1,\nu_2},\] the F distribution with $$(\nu_1,\nu_2)$$ degrees of freedom. This distribution is used, for example, for inference about the ratio of two variances, in Analysis of Variance (ANOVA) and in simple and multiple linear regression.