Pseudo-scalar product on Manifold

Question

I'm trying to study the Semi-Riemannian Manifold and the relativity (I use the book Semi-Riemannian Manifold- O'Neill). But I don't understand the following thing: In a Semi-Riemannnian Manifold, I can define a pseudo-scalar product on each tangent space $T_pM$ with the metric tensor $g$. But whyI have to define it on the tangent space? And if I define it on the tangent spaces, I have automatically define a pseudo-scalar product on all the manifold? If I think at a space-time, for example $\mathbb{M}^4$, the Minkosky space time, i need only the scalar product on the tangent space to know if a curve is spacelike or timelike? Sorry but I'm really confused. Thanks