the tangent space
Stand at one point on a curved surface and ask: what are all the directions and speeds in which I could set off from here? On a smooth ball, the honest answer is a flat plane of arrows touching the ball at that point — the tangent plane. The tangent space generalizes this to any manifold and any dimension: at each point p it is the flat vector space of all possible velocity vectors of curves passing through p. It is the best flat approximation to the manifold right at p, the linear world in which calculus happens.
There are two equivalent ways to make this precise. The picture way: take any smooth curve through p; its velocity at p is a tangent vector, and the tangent space T_p M is the set of all such velocities. The algebra way, which frees us from any ambient space: a tangent vector is a derivation — a rule that takes any smooth function near p and returns a number (its rate of change in that direction), obeying linearity and the product rule f g -> f(p)(rate of g) + g(p)(rate of f). Both definitions yield the same n-dimensional vector space. In a chart with coordinates x^1, ..., x^n, the partial-derivative operators d/dx^1, ..., d/dx^n form a basis, so every tangent vector is a combination v = v^1 d/dx^1 + ... + v^n d/dx^n.
The tangent space is where the metric will later live (it measures lengths of these velocity vectors and angles between them) and where vector fields, geodesics, and curvature are all built. A vital subtlety: tangent spaces at two different points are genuinely different spaces — there is no automatic way to compare a vector at p with a vector at a faraway point q. Supplying such a comparison is exactly the job of a connection and parallel transport; without one, 'the same direction over there' has no meaning on a curved manifold.
Take the unit circle in the plane and the point p = (1, 0). A curve running along the circle through p, say (cos t, sin t), has velocity at t = 0 equal to (0, 1). Every velocity of every curve through p is a multiple of (0, 1), so the tangent space at p is the vertical line of vectors {(0, c)} — a 1-dimensional flat space touching the circle at p.
The tangent space collects every possible velocity at a point; for the circle it is just a line.
Do not imagine the tangent space as floating physically off the surface — that picture only works for shapes drawn inside R^3. Intrinsically it is an abstract vector space attached to the point, defined entirely by directional derivatives.