The backbone of Connes' approach to noncommutative geometry is the ChernConnes pairing. By discovering the cyclic homology of an algebra and then pairing it the $K$theory of that algebra, Connes showed how to find numerical invariants of $K$theory classes. This is analaogous to the way in which the Chern character map gives numerical invariants of vector bundles. I would like to know what has the ChernConnes pairing been used for? Besides its intrinsic interest as a noncommutative version of a classical phenomenon, why is it important?
The Selberg principle in representation theory of semisimple $p$adic groups $G$, says that the orbital integral of a coefficient of supercuspidal representation of $G$, on a hyperbolic conjugacy class in $G$, is $0$. Connes suggested around 1982 that this might be interpreted by means of the ChernConnes pairing: indeed a coefficient of supercuspidal representation is, up to a scalar factor, an idempotent in the convolution algebra $C_c^\infty(G)$, while an orbital orbital gives a trace over $C_c^\infty(G)$; hence the interpretation of the Selberg principle by saying that certain traces give the zero pairing on the $K$theory $K_0(C_c^\infty(G))$. This program was implemented for rank 1 groups in this paper: http://www.theta.ro/jot/archive/1986016002/1986016002007.pdf 


I guess you named already one if the biggest points yourself: getting invariants! So more specifically, Connes obtained (with Moscovici and others) invariants of pretty ugly foliations, it helps in the classifying certain $C^*$algebras etc. There are also more algebraic versions in deformation quantization, where people have formulated index theorems using this pairing. In some sense, it is also a new way of understanding index theorems in general, though I must admit that I'm not really an expert here... 

