Tangent Space Question (Differential Topology)

This is more of a conceptual question. Given some manifolds and smooth mapping . Then I realize for an the tangent space at the point can be visualized as the hyperplane touching at that point and then translated to the origin, but what point are we associating to the origin. Is it the vector x that get's translated to the origin and similarly f(x) for . Sorry if this question isn't worded properly or too trivial, I'm reading Milner Diff. Topology and though excellent, it's a bit vague at times.

Thanks.