You need to understand that differential is a linear map in the tangent plane. Note that a smooth function on R^n is a smooth map between two manifolds R^n and R. So if you can understand differential of a general smooth map, it will helps you to understand differential of a smooth function, which is a real-valued map.

Take an example, suppose S^n is the n-dimensional sphere, and f: S^n -> S^n is the anti-podal map f(p) = -p. How do you define its differential?

See

Differential of a function - Wikipedia, the free encyclopedia for more to understand that differential is a map.