From the point of view of coordinate charts, the notion of tangent space is quite simple. The tangent space consists of all directions, or velocities, a particle can take. In an open set U in R^n there are no constraints, so the tangent space at a point p is another copy of R^n. The set U could be a coordinate chart for an n-dimensional manifold.

calculus | coordinate chart | differential k-form | directional derivative | Euclidean space | exterior algebra | Jacobian | manifold | submanifold | tangent bundle | tangent space | vector field | velocity vector