According to the book Linear algebra and its applications by Strang, "(The) good method is Gaussian Elimination. This is the algorithm that is constantly used to solve large systems of equations".
Is it really constantly used to solve large systems of equations, or are there better algorithms for it? If yes, which algorithms are usually used to actually solve them?
It depends on what you mean by large. On most regular computers you can solve dense systems up to 5k x 5k fairly easily. If the matrix is ill-conditioned you can use other algorithms and if it pretty large it is probably better to use iterative methods.