What is Weighted and generalised least squares?

Weighted least squares minimises $\sum_iw_i(y_i - x_i^\top\beta)^2$, with weights inversely proportional to the variance of each observation’s error. Generalised least squares minimises $(y - X\beta)^\top\Omega^{-1}(y - X\beta)$ for an error covariance $\Omega$; it is OLS after multiplying the model by $\Omega^{-1/2}$, and it is the efficient linear unbiased estimator when $\Omega$ is known.