There is nothing unique to SPSS how the sums of squares are computed. The formulas for computing (Type III and other types) of sums of squares are in the GLM algorithms, though it takes some work to dig things out from multiple places and you have to understand the canonical overparameterized model used by GLM in order to make sense of how the L matrices are formed and applied.
In SPSS, the default SS computation is Type III (so-called "regression" sum of squares; years ago, SPSS called this "unique" or "simultaneous" SS). This solution is that which most other statistics software would use or default to.
If you have a balanced (equal cell n) design, Type I, Type II, and Type III SS are identical. For unbalanced designs, Type I will estimate main effects first, in order of entry in the GLM command, then subsequent interactions controlling for previously estimated effects; Type II will estimate main effects first, controlling for each other, then interactions controlling for previously estimated effects; Type III estimates all effects simultaneously, controlling for each other. Type IV is a variant of Type III, which can be useful when you have missing cells in a multi-factor design.
Here's a couple of links that address whys for each: