Conceptual undestanding of a pull-back in differential geometry I have recently started learning a bit of differential geometry and have reached the point of learning about diffeomorphisms and pull-backs.
In the notes that I've been reading the author states that:
$``$Let $f: M\rightarrow\mathbb{R}$. A diffeomorphism $\phi :M\rightarrow M$ changes $f$ into a different function, called the pull-back of $f$, and denoted as $\phi^{\ast}f$. It is defined such that given two points $p\in M$ and $q=\phi (p)\in M$, $$ (\phi^{\ast}f)(p)=f(q)=f(\phi (p))$$ i.e. $$(\phi^{\ast}f)=f\circ\phi"$$  
By this, is it meant that the pull-back function evaluated at the original point $p\in M$ is equal to the "original" function evaluated at the point $q\in M$ mapped to by the diffeomorphism? 
Is the reason for defining such a notion so that one can compare the values of a function evaluated at different points on the manifold?  
 A: $\newcommand{\Reals}{\mathbf{R}}\newcommand{\Smooth}[1]{\mathscr{C}^{\infty}(#1)}$Your understanding of the definition of $\phi^{*}f$ is perfectly correct. As for the "reason", it's probably more productive to ask "how do pullbacks arise in theory and practice", and to treat each "satisfying occurrence" as a "reason". :)
I Am Not A Historian of Mathematics, but probably the origin of the definition was the wish to parametrize a piece of a manifold, i.e., to look at diffeomorphisms $\phi^{-1}:\phi(U) \to U$ (with $\phi: U \subset M \to \Reals^{n}$ a coordinate chart on $M$), then to express questions about calculus on $M$ as questions about ordinary multivariable calculus in the open set $\phi(U) \subset \Reals^{n}$.
Another reason (though probably a secondary or tertiary reason) is to study what parts of differential and integral calculus are diffeomorphism-invariant, a.k.a., independent of the choice of coordinate system. Examples include velocites of curves (but not accelerations, which require additional structure, such as a Riemannian metric), and exterior calculus (differential forms, Stokes' theorem). If you're looking for an additional reference, Spivak's Comprehensive Introduction to Differential Geometry, Volume I, is quite good. (I believe there are many excellent newer books, as well; my bibliographic knowledge is rather out-of-date.)
A more modern "reason" for looking at pullbacks is the prospect of replacing a smooth manifold (or a locally-compact Hausdorff space) by its algebra $\Smooth{M}$ of smooth functions. As you've noted, a diffeomorphism $\phi:M \to M$ induces an automorphism of the algebra $\Smooth{M}$. A point $p$ of $M$ corresponds in this picture to the maximal ideal of functions vanishing at $p$. Many constructs of differential geometry can be formulated similarly in terms of algebras on $M$. This idea is Deep, giving rise to the scheme-theoretic viewpoint in algebraic geometry, and to non-commutative geometry.
Incidentally, the definition of pullback of a function makes perfect sense for a smooth mapping of manifolds $\phi:M \to N$, not merely for diffeomorphisms $\phi:M \to M$. Generally, the pullback $\phi^{*}:\Smooth{N} \to \Smooth{M}$ isn't invertible. (If $\phi$ is a constant mapping, for example, the pullback amounts to evaluation.) Pullback by a diffeomorphism, however, is invertible; the inverse is $(\phi^{-1})^{*}:\Smooth{M} \to \Smooth{N}$, pullback by the inverse map.
A: Yes to 'By this...' and what follows. 
I guess the reason for defining such a notion is first of all, that $\phi$ defines a a natural map from $C^\infty(M)$ to $C^\infty(M)$ for which one wants have a name, and which has certain formal properties which behave well from a functiorial point of view (the concept is not limited to the situation you are looking at).
As a first generalization consider a diffeomorphism $\phi: M\rightarrow N$. The you get a pullback $\phi:C^\infty(N)\rightarrow C^\infty(M)$ by the same definition.
