Projective space is supposed to parametrize lines through the origin. A line is determined by two points, so a line through the origin is determined by any nonzero vector.
As Nate's explains, you can certainly include 0, but you will get a different space.
The reasons we care to parametrize lines through the origin is deep (by hich I mean supported by many nontrivial theorems and examples). One answer is that projective space is a more natural setting for (algebraic) geometry than affine space, in the sense that theorems fewer less special cases (Bezout's theorem or the classification of plane conics, 27 lines on a cubic, etc.). We can think of projective space as a natural compactification of affine space, which is designed to catch points that wander off to infinity by assigning to their limit the direction they wandered off in.
(I know this is an old question, but maybe somebody has a similar question, and I think this answer is sufficiently different from the others...)