the 4th colum for 3D transformation matrix, is to do translation. For other transformations like rotation, scaling etc all that is needed is multiplication e.g. Scale
x''=x * sX
y''=y * sY
z''=z * sZ

but a translation consists of a addition ( if value is negative a subtraction ) and because of the mechanics of matrix multiplication they can''t be applied. So the 4th column is used for translation. The 4th componet of a vector is set to 1 to allow translation.Ill give a example of 2D vectors & therfore 3x3 matrices of translation. Ill translate the vector by (4,2)

(2,3,1) x (1,0,0)
(0,1,0)
(4,2,1)

x'' = (2 * 1 ) + (0 * 3) + (1 * 4 ) = 6 // correct
y'' = (0 * 2 ) + (1 * 3) + (1 * 2 ) = 5 // correct

the next column doesn''t really matter unless your doing some sort of projection matrix. The important part of the example is the brackets at the end which actully do the translation

hope this helped
Iain