Ordinary least squares treats all residuals as equally informative. In experiments, measurements often have different known standard uncertainties $\sigma_i$. Then a residual of $0.1$ may be negligible for a noisy point with $\sigma=1$ but serious for a precise point with $\sigma=0.01$.

For independent approximately Gaussian measurement errors, weighted least squares minimizes $$\chi^2(\theta)=\sum_{i=1}^n\frac{[y_i-f(x_i;\theta)]^2}{\sigma_i^2}.$$ Equivalently, each point receives weight $w_i=1/\sigma_i^2$. More precise observations influence the fit more strongly because their deviations are less plausibly explained by measurement noise.

Consider estimating a constant $\mu$ from two measurements: $10.0\pm1.0$ and $12.0\pm0.5$. The weighted estimate is $$\hat\mu=\frac{10/1^2+12/0.5^2}{1/1^2+1/0.5^2} =\frac{10+48}{1+4}=11.6.$$ The second measurement pulls the result closer to $12$ because it has four times the weight.

If measurement errors are correlated, a covariance matrix $\Sigma$ replaces independent weights: $$\chi^2=(\mathbf y-\mathbf f)^{\mathsf T}\Sigma^{-1}(\mathbf y-\mathbf f).$$ Ignoring correlation can make a fit appear to contain more independent information than it actually does.

Weights should represent the measurement model, not be chosen afterward to make a curve look better. If the quoted uncertainties themselves depend on fitted parameters or the noise is strongly non-Gaussian, a likelihood model may be more appropriate. Weighted least squares is therefore best understood as a statistical consequence of a particular uncertainty model, not as a cosmetic fitting option.