The 'Augmented Matrix' is a matrix representing a system of simultaneous linear equations. It is called an augmented matrix because the matrix of coefficients is augmented by an extra column representing the constants to which each equation in equal to in turn.
Suppose we have the set of simultaneous linear equations
The matrix of coefficients is
and this is augmented by the column of constants from the right hand side,
The result is the matrix
The vertical line drawn after the entires of the coefficient matrix indicated that the matrix is augmented by the entries on the right hand side of this line.
The augmented matrix is used to solve the associated system by performing a series of 'elementary row operations' on it. These elementary row operations may be any of
Interchanging two rows
Multiplying or dividing every element in a row by a number
Adding a multiple of a row to another row