A good introduction to the basics.

What is also worth pointing out and which was somewhat glanced over is the close connection between the weight function and the polynomials. For different weight functions you get different classes of orthogonal polynomials. Orthogonal has to be understood in relation to the scalar product given by integrating with respect to the weight function as well.

Interestingly Gauss-Hermite integrates on the entire real line, so from -infinity to infinity. So the choice of weight function also influences the choice of integration domain.