tangent space
You have your differentiable manifold \(M\). You can attach a tangent space \(T_x M\) to every point \(x \in M\). The collection of all such spaces is called \(TM\).
1. submanifolds
If you have a submanifold and your \((n-1)\) dimensional surface is defined by a function, you can define a tangent space by taking a linear approximation of that function, i.e., a derivative.
2. intrinsic
The intrinsic way to talk about tangent spaces is that each member of the tangent space is an equivalence class of curves \(\gamma\) on \(M\). All curves in the same class have the same first derivative.