What is Gaussian Quadrature?

Gaussian quadrature is a method that replaces an integral by a sum. As an example, consider the following 1D integral:

Gaussian quadrature numerically integrates f(s) using a sum,

In order to perform numerical integration, Gaussian quadrature requires that the Order of f(s) = 2 x NQP - 1. Hence for a linear function (order 1), (1 + 1) / 2 = 1 NQP is required. For a cubic function (order 3), (3 + 1) / 2 = 2 NQPs are required to integrate the function exactly.

Gauss points for the 1x1, 2x2 and 3x3 integration schemes when applied to FEM

Also read
The role of Gaussian quadrature in deriving local stiffness matrix (k)