Paul's Online Math Notes at Lamar University offers definitions and instructions on solving augmented matrices. It goes through several operations that can be used to manipulate augmented matrices to help solve a system of equations.
Paul's Online Math Notes describes how to convert a system of equations into an augmented matrix, moving the coefficients of each variable in each equation into the left side of the matrix, and moving the constants on the right side. It then defines some matrix operations that can manipulate the augmented matrix, such as interchanging rows, multiplying a row by a constant and adding one row to another row. It then uses these operations to go through a process called Gauss-Jordan elimination to solve an augmented matrix. The intent of Gauss-Jordan elimination is to produce the identity matrix on the left side of the augmented matrix and the solutions to the equation on the right side.
Purplemath also offers a page explaining the basics of augmented matrices and converting a linear system to an augmented matrix. Tutorvista provides several examples on how to solve simple augmented matrices when dealing with just two variables.