A quadrature rule with $n$ sample points has freedom to choose both its nodes and its weights. Gaussian quadrature chooses them so that polynomial exactness is as high as possible.

For ordinary Gauss-Legendre quadrature on $[-1,1]$, an $n$-point rule has the form

$$\int_{-1}^{1} f(x),dx\approx\sum_{i=1}^{n} w_i f(x_i),$$

with specially chosen nodes $x_i$ and weights $w_i$.

The $n$-point rule integrates every polynomial of degree up to

$$2n-1$$

exactly.

The nodes are not equally spaced and generally do not include the interval endpoints. An integral over another finite interval can be mapped to $[-1,1]$ by a change of variables.

Gaussian quadrature is especially effective when evaluating smooth integrands is expensive, and it is widely used when assembling finite-element integrals.