It's worth keeping in mind here that the 1979 paper by Kazhdan and Lusztig dealt quite generally with Iwahori-Hecke algebras of arbitrary Coxeter groups, not just the finite Weyl groups (or even the Weyl groups attached to Kac-Moody algebras). For the finite and affine Weyl groups the connections with Schubert varieties are a major motivation for the Kazhdan-Lusztig polynomials, since the polynomials encode important geometric data as Leandro points out.
For the types of representation theory that involve Coxeter groups in the role of "Weyl groups", there tend to be good analogues of the original conjecture for finite dimensional semisimple Lie algebras and their highest weight representations of arbitrary dimension. This is certainly a major application of the ideas, since it leads to new alternating sum formulas for unknown characters, with values of the polynomials at 1 as the coefficients.
On the other hand, it's essential to build into this kind of character formula the pair of elements of the Coxeter group over which summation occurs: these are related by the Chevalley-Bruhat ordering, also a very general notion for Coxeter groups. In fact, the polynomials in isolation are not especially noteworthy. As Patrick Polo showed in type A, any non-negative polynomial with integral coefficients and constant term 1 shows up somewhere as a Kazhdan-Lusztig polynomial. (The non-negativity of coefficients is still an intriguing open conjecture from the original K-L paper, with no apparent general interpretation of the polynomials as a guide.)
I should also mention that the coefficients of the polynomials associated to a finite Weyl group determine the multiplicities of simple modules in layers of the Jantzen filtration for a Verma module, as indicated by Chuck in a wider setting. However, the proof by Beilinson-Bernstein of Jantzen's older conjecture on the filtrations shows that his conjecture is even stronger than the Kazhdan-Lusztig conjecture. (Chapter 8 of my graduate text surveys a lot of this in detail, but without being a complete treatment.)