Differential structure allows one to define the globally differentiable tangent space, differentiable functions, and differentiable tensor and vector fields. Differentiable manifolds are very important in physics
Yes, but I don't understand which method is used, I need to see an example.
How is a tensor defined on the tangent space in the point associated with each point of a differentiable manifold, for example an open of the Euclidean space $R^n%$?
We can define a vector field on a differentiable manifold, where the vector at each point $\mathbf{P}$ is an element of the tangent space $\mathbf{T_P}$ at $\mathbf{P}$.
As well as a tangent space $\mathbf{T_P}$, there is also a cotangent space $\mathbf{T^*_P}$ at each point $\mathbf{P}$. This is the space of linear maps from $\mathbf{T_P}$ to $\mathbb{R}$ (or from $\mathbf{T_P}$ to $\mathbb{C}$ if we are considering a complex manifold).
Once we have a tangent and cotangent space at each point then we can define a tensor field where the tensor at each point is a linear map from some product of copies of $\mathbf{T_P}$ and $\mathbf{T^*_P}$ to $\mathbb{R}$ (or $\mathbb{C}$ ). A tensor field may represent some intrinsic attributes of the manifold (the Riemann curvature tensor field, for example) or it may represent extrinsic physical attributes (the stress-energy tensor field, for example).