I've recently become somewhat interested in smooth manifolds and surfaces as a result of preparation for various interviews, amongst other reasons. The concept of a surface is very intuitive, and is a concept that a student of mathematics is likely to encounter very early in their mathematical careers, though a reasonable definition of a surface takes more effort. The result of this formalisation is a remarkably elegant theory, which eventually leads to ideas of smooth manifolds - arguably one of the best sounding mathematical objects - and generalised calculus.
It took quite some time before I had a "big picture" of what a surface is, and how this fits into the grander theory. Undoubtedly I am missing some of the major pieces to this puzzle, but it does show some of the elegance of the theory (at least for me).
Let us return to the relatively basic theory of differential calculus. Our first introduction to differentiation usually comes during A-level maths, where we are told various standard derivatives without much explanation, and then that these derivatives represent the gradient of a curve at a given point. (For those who may need to look at derivatives once again, I suggest the Wikipedia page on differentiation, which has some lovely illustrations that will help with this discussion.)
Geometrically, the what is happening is that we are constructing a line that meets the curve only once at our selected point (here we only consider the points that are relatively "close-by", a tangent line might cross the curve elsewhere but this should be sufficiently "far away"). The gradient of this tangent line is equal to the value of the derivative at the chosen point.
Lines are very simple geometric figures that we can easily describe and manipulate. If we start at our points and move along the tangent line a small amount, then the point on the tangent line will be very close to the point on the curve a similar distance from our selected point. We might say then that a derivative gives us the means of approximating the value of a complicated curve using straight lines nearby to a given point. (In fact, this is the fundamental idea that underpins a large number of numerical methods for solving differential equations.)
In three dimensions, we have a new possibilities to consider. We are no longer restricted to curves, and we can instead investigate the properties of surfaces such as the unit sphere (those points that have distance 1 to the origin). Here we can no longer approximate using a single line, since the sphere expands away from any given point in many directions. This is also reflected in the equation that determines the points in the sphere, which has two independent variables. This is typical of surfaces in three-dimensional space. (Think of a sheet that has peaks and valleys, you can move about on the sheet as if you were in the plane, although your movements also move you through the third, unseen dimension.)
Many surfaces can be realised as the zero set of some function of several variables. The partial derivatives of such a function give us information about how the function evolves in the direction of each of its variables, which are usually the x, y, z (and so on) directions. From these partial derivatives, we can find a directional derivative of our function in any direction by taking a weighted combination of the partial derivatives (a linear combination).
The evolution of a surface, as we move away from a given point, can be approximated by the set of all the possible weighted combinations of the partial derivatives at that point. This is the tangent space of the surface at the point. In the case of the unit sphere, the associated function has two variables so the two partial derivatives determine a plane at each point.
Tangent spaces provide an essential tool in differential geometry - the study of smooth surfaces - because it is much easier to understand a plane than a complex surface.