If you have
[a b][e f]
[c d][g h]
when you multiply you get
[(a*e+b*g) (a*f+b*h)]
[(c*e+c*g) (c*f+d*h)]
Basically to get the first row and the first column (ie the first entry) you multiply and add the first row of the first matrix (a and b) by the first column of the second matrix (e and g). For the first row and second column, you use the first row of the first matrix (a and b) and the second column of the second matrix (f and h), etc. You do this for every entry of the resulting matrix.
So if we have
[1 5][4 -2]
[-2 3][0 1]
it comes to
[(1*4+5*0) (1*-2+5*1)]
[(-2*4+3*0) (-2*-2+3*1)]
[(4+0) (-2+5)]
[(-8+0) (4+3)]
[4 3]
[-8 7]