What is the significance of non-commutative geometry in mathematics?

I think I'm in a pretty good position to answer this question because I am a graduate student working in noncommutative geometry who entered the subject a little bit skeptical about its relevance to the rest of mathematics. To this day I sometimes find it hard to get excited about purely "noncommutative" results, but the subject has its tentacles in so many other areas that I never get bored.

Before saying anything further, I need to say a few words about the Atiyah-Singer index theorem. This theorem asserts that if $D$ is an elliptic differential operator on a manifold $M$ then its Fredholm index $dim(ker(D)) - dim(coker(D))$ can be computed by integrating certain characteristic classes of $M$. Non-trivial corollaries (obtained by "plugging in" well-chosen differential operators) include the generalized Gauss-Bonnet formula, the Hirzebruch signature theorem, and the Hirzebruch-Riemann-Roch formula. It was quickly realized (first by Atiyah, I think) that the proof of the theorem can be viewed as a statement about the Poincare duality pairing between topological K-theory and its associated homology theory (these days called K-homology).

I wasn't around, but I'm told that people were very excited about Atiyah and Singer's achievement (understandably so!) People quickly began trying to generalize and strengthen the theorem, and my claim is that noncommutative geometry is the area of mathematics that emerged from these attempts. Saying that marginalizes the other important reasons for developing the subject, but I think it was Connes' main motivation and in any event it is a convenient oversimplification for a MO answer. It also helps me answer your first question by playing to my personal biases: when I was choosing an area of research I told my adviser that I was interested in learning more about that Atiyah-Singer index theorem and I was led inexorably toward the tools of noncommutative geometry.

The origin of the relationship between NCG and Atiyah-Singer lies in equivariant index theory. Atiyah and Singer realized from the start that if $M$ admits an action by a compact Lie group $G$ and $D$ is invariant under the group action then it is better to think of the index of $D$ as a virtual representation of $G$ (i.e. an element of the $G$-equivariant K-theory of a point) rather than as an integer. If $G$ is not compact then this doesn't really work, but the noncommutative geometers realized that $D$ does have an index in the K-theory of the reduced group C$^\ast$-algebra $C_r^\ast(G)$. Indeed, to a noncommutative geometer equivariant index theory is all about a map $K_\ast(M) \to K_\ast(C_r^\ast(M)$ where $K_\ast(M)$ is the K-homology of $M$; in the case where $M$ is the universal classifying space of $G$, Baum and Connes conjectured that this map is an isomorphism. Proving this conjecture for more and more groups and understanding its consequences motivates a great deal of the development of NCG to this day.

The conjecture is interesting in its own right if you already care about index theory, but even if you don't injectivity of the Baum-Connes map implies the Novikov conjecture (see Alain Valette's answer) and surjectivity is related to the Borel conjecture. It has numerous other applications, for example to the theory of positive scalar curvature obstructions in Riemannian geometry or to the Kadison-Kaplansky conjecture in functional analysis (which would follow from surjectivity). Recently there has been a lot of interest in connections between the Baum-Connes conjecture and representation theory; the Baum-Aubert-Plyman conjecture in p-adic representation theory has its origins in these sorts of considerations.

Much of the rest of NCG can also be traced back to index theory. Kasparov's KK-theory arose as a way to understand maps and pairings between K-theory and K-homology, motivated in part by index theory. Connes' work on noncommutative measure theory arose from his work on index theory for measurable foliations (with applications to dynamical systems). Cyclic (co)homology was invented in part to gain access in a noncommutative setting to the Chern character map from K-theory to cohomology which translates the K-theoretic formulation of the index theorem into a cohomological formula. Connes' theory of spectral triples and noncommutative Riemannian geometry is based on the theory of Dirac operators which was invented by Atiyah and Singer to prove the index theorem. I guess my point with all of this is that all the esoteric machinery of NCG seems less artificial when viewed through the lens of index theory.

My favorite example concerns Novikov conjecture, on the homotopy invariance of higher signatures for closed manifolds with fundamental group $G$: see http://en.wikipedia.org/wiki/Novikov_conjecture (note that this Wikipedia entry rather stupidly says that it has been proved for finitely generated abelian groups: that's correct, but it was proved for MANY more groups, e.g. hyperbolic groups, countable subgroups of $GL_n(\mathbb{C})$, etc...). I think we agree that this is a conjecture in topology.

Now, look at this remarkable result by Guoliang Yu ( http://www.kryakin.com/files/Invent_mat_(2_8)/139/139_21.pdf )

"If the group $G$ admits a coarse embedding into Hilbert space, then it satisfies the Novikov conjecture".

A coarse embedding is a map $f:G\rightarrow L^2$ for which there exists control functions $\rho_{\pm}:\mathbb{R}^+\rightarrow\mathbb{R}$, with $\lim_{t\rightarrow\infty}\rho_\pm(t)=\infty$, which ``control'' $f$ in the sense hat, for every $x,y\in G$:

$$\rho_-(|x^{-1}y|_S)\leq\|f(x)-f(y)\|_2\leq \rho_+(|x^{-1}y|_S),$$ where $|.|_S$ denotes word length with respect to some finite generating subset $S$ in $G$. The existence of a coarse embedding is a weak metric condition (actually we know of basically just one class of groups which do not admit such an embedding, the ``Gromov monsters''). And this weak metric condition, quite surprisingly, implies a strong consequence in topology.

Now my point is that the two known proofs of Yu's result (the original one, and the one by Skandalis-Tu-Yu, see http://www.math.univ-metz.fr/~tu/publi/coarse.pdf) both appeal in a fundamental way to the tools of non-commutative geometry: $C^*$-algebras, $K$-theory, groupoids, Kasparov's $KK$-theory (to be precise: ``equivariant $KK$-theory for groupoids'').

Now to answer your first question: how to motivate a graduate student? Well, the subject mixes classical geometry, algebraic topology, non-commutative algebra, functional analysis, so it is one of those subjects that give you a feeling of the unity of mathematics...

There are much better answers above than this one, but:

If you believe fiber bundles are important to classical mathematics, then you probably believe fibrations are, and maybe foliations are, as well. If you don't, note that a foliation of a smooth manifold is a decomposition of the manifold into integral submanifolds (roughly, solutions to differential equations). You can't get much more classical than this. In his book Noncommutative Geometry Connes tried to make it clear that to understand the leaf space of a foliation, more is needed than the classical quotient construction, groupoids and noncommutative geometry give more information about a patently classical "space". You probably say: So what? There are other ways. Connes tries then to show us that there is a connection between a fundamental von Neumann algebra invariant (the flow of weights) and one of the key invariants for a codimension 1 foliation (the Godbillon-Vey class), which appears in the first chapter on many introductory accounts of foliations. I find it hard to believe that this is coincidental. For me, this warrants closer investigation.

The index theorem for measured foliations discussed above perhaps grew from a seed like the above mentioned connection. (I wonder what we need to do to get Connes to weigh-in over here at MO?)