Basics
(defns) 

Humphreys 
Reflection groups and coxeter groups Shields Library QA171 .H833 1990 Regular Loan 
Ch 7, pg 145; introductory
lectures and basic definitions will largely follow this text. If you
are rusty on the idea of coxeter groups and length, read here (Ch 1, Ch
5), and also try HW1. 
Andersen, Jantzen, Soergel.  Representations
of quantum groups at ApTH root of unity and of semisimple groups in
characteristic p, independence of p Shields Library QA176 .A53 1994 Regular Loan 

Andersen 
The irreducible characters for
semisimple algebraic groups and for quantum groups, Proc. ICM94 (Vol II, 1995), 731743. 
see section 4 
Brenti 
KazhdanLusztig polynomials: history
problems, and combinatorial invariance. Sém. Lothar.
Combin. 49 (2002/04), Art. B49b, 30 pp. (electronic). 
Good intro, defn's (a more
combinatorical POV than Humphreys) 
Ariki 
Lectures on cyclotomic Hecke
algebras; http://arxiv.org/abs/math.QA/9908005 
a nice overview 
Parabolic KL polys 

Frenkel, Khovanov,
Kirillov 
KazhdanLusztig polynomials and
canonical basis http://xxx.lanl.gov/abs/qalg/9709042 
A good place for basic defns of
parabolic KL polys, and also very handon feel for quantum groups and
canonical bases in a certain repr. In this paper they show that the KazhdanLusztig polynomials (and, more generally, parabolic KL polynomials) for the group $S_n$ coincide with the coefficients of the canonical basis in $n$th tensor power of the fundamental representation of the quantum group $U_q sl_k$. 
Leclerc, Thibon 
LittlewoodRichardson
coefficients and KazhdanLusztig polynomials; http://arxiv.org/ps/math.QA/9809122 
good exposition! They show
that the LittlewoodRichardson coefficients are values at 1 of certain
parabolic KazhdanLusztig polynomials for affine symmetric groups. 
Deodhar 
On some geometric aspects of Bruhat orderings II. The parabolic analogue of KazhdanLusztig polynomials, Journal of Algebra 111 (1987), 483506. 
this might be where parabolic KLs
are first defined. 
computation
and combinatorics


Brenti 
KazhdanLusztig and
$R$polynomials, Young's lattice, and Dyck partitions.
Pacific J. Math. 207 (2002), no. 2, 257286. 

Brenti 
many other papers 
Brenti has many papers on
combinatorial properties of KL polys and using them for
computation. Combinatorics of posets come in. as to qanalogues
of fibonacci numbers, and things about increasing
subsequences. If interested, let me know and i will expand
on this list. 
Lascoux; Schützenberger 
Polynômes de Kazhdan & Lusztig pour les grassmanniennes. (French) [KazhdanLusztig polynomials for Grassmannians] 
does anyone read french? this is if you are interested in the geometry, and what KL polys have to do with intersection cohomology of schubert cells. 
Relating
to canonical bases/ Fock space 

Leclerc 
Decomposition numbers and
canonical bases; http://arxiv.org/abs/math.QA/9902006 
They obtain some simple
relations between decomposition numbers of quantized Schur algebras at
an nth root of unity (over a field of characteristic 0). These
relations imply that every decomposition number for such an algebra
occurs as a decomposition number for some Hecke algebra of type A. We
prove similar relations between coefficients of the canonical basis of
the qdeformed Fock space previously introduced in a joint work with
Thibon. [nice summary of history] 
(LLT) Lascoux, Leclerc, Thibon  Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), 205263.  They give
the algorithm for computing the global crystal basis
of the basic representation of a quantum affine algebra of type A_{n}^{(1)}.
They conjecture [since proved by Ariki, Grojnowski] that at
q=1 , they get decomposition numbers for Hecke
algebras at n th root of 1. Very concrete
description of Fock space and action. [you might find cleaner
exposition in later papers] 
Goodman  Fock
space and KL polys 
some very nice lecture notes. 
Goodman and Wenzl  A path algorithm for affine KazhdanLusztig polynomials, Math. Z. 237 (2001), no. 2, 235249. Preprint version at math.RT/0011245  They develop an algorithm for
computing affine KazhdanLusztig polynomials, for all Lie types. This
generalizes our previously published algorithm for type A (see paper
below),
which in turn is a faster version of an algorithm due to Lascouz,
Leclerc and
Thibon. (there is some relation to Littelmann paths) 
Goodman and Wenzl  Crystal bases of quantum affine algebras and affine KazhdanLusztig polynomials. Internat. Math. Res. Notices 1999, no. 5, 251275. Preprint version at math.QA/9807014  This is type A. They
show that the coefficients of the lower global crystal base
for the Fock representation of quantum affine
sl_n coincide with certain affine KazhdanLusztig polynomials. These
are qanalogues of decomposition
numbers for Specht modules of the Hecke algebra of type A_n; the
coefficients of the affine KazhdanLusztig polynomials are qanalogues
of
decomposition numbers for tilting modules for quantum sl_k. Their
algorithm
allows fast computation of these decomposition numbers, improving on
the algorithm of Lascoux, Leclerc, and Thibon
for the lower global crystal base . Students interested in algorithms, computation, complexity, might like to read this. Also you get a very handson feel for the Fock space. 
Goodman and Wenzl  IwahoriHecke algebras of type $A$ at roots of unity. J. Algebra 215 (1999), no. 2, 694734. Preprint version at qalg/9610033  
Mark Goresky  TABLES OF KAZHDANLUSZTIG POLYNOMIALS  once we've gone through the basic def'ns, you may want to look here to see examples 
Geometry 

Varagnolo and Vasserot 
On the decomposition matrices of the quantized Schur algebra
http://www.arxiv.org/abs/math.QA/9803023 

Kashiwara, Saito 
Geometric Construction of
Crystal Bases; Research Institute for Mathematical Sciences, Kyoto University, preprint RIMS1081 
The result is very geometric,
but the intro is a good read. They realize a crystal as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety. 
Kashiwara, Tanisaki 
Parabolic KazhdanLusztig
polynomials and Schubert varieties math.QA/9809122 
They give a
description of the intersection cohomology groups of the Schubert
varieties in partial flag manifolds over symmetrizable KacMoody Lie
algebras in terms of parabolic KazhdanLusztig polynomials introduced
by Deodhar. 
Billey, Warrington  KazhdanLusztig polynomials for 321hexagonavoiding permutations; http://arxiv.org/abs/math.CO/0005052  This is a mixture of geometry
and combinatorics. It uses the Schubert
point of view. If you are interested in this, Billey has several
other
papers along these lines. They give a combinatorial formula for the KazhdanLusztig polynomials $P_{x,w}$ in the symmetric group when $w$ is a 321hexagonavoiding permutation. 
Immanants 

Haiman 
Hecke algebra characters and
immanant conjectures J. Amer. Math. Soc. 6 (1993) no. 3, 569595. PostScript, PDF 
Two conjectures on characters of the Hecke algebra of type A_{n}, evaluated on KazhdanLusztig basis elements. Theorem: immanants of JacobiTrudi matrices are positive combinations of Schur functions. If Conjecture 1 holds, then "monomial" immanants are also Schur positive. 
Rhoades , Skandera 
KazhdanLusztig immanants  this and the paper above are a
very different sort of way to study the KL basis and symmetric functions 
Cells 

Kazhdan and Lusztig 
Representations of Coxeter groups and Hecke algebras; Invent. Math. 53 (1979), no. 2, 165  184 
This is where KL polys are defined,
as is the relation on W by which right, left, 2sided cells are defined. 
Graham ; Lehrer 
Cellular algebras,
Invent. Math. 123 (1996), 1–34. 
more details on cells and the
resulting representations 
LUSZTIG 
Left cells in Weyl groups, Lie Group Representations, I (R. L. R. Herb and J. Rosenberg, eds.), Lecture Notes in Math., vol. 1024, SpringerVerlag, 1983, pp. 99111. or Cells in affine Weyl groups, Advanced Studies in Pure Math. 6, Algebraic groups and related topics, Kinokuniya and NorthHolland, 1985, 255287. or Lectures on affine Hecke algebras with unequal parameters. Available at arXiv:math.RT/0108172. 
More about cells. In type A:
When you look at the w
in a certain left cell that defines a Specht module of shape lambda,
all those w give a pair P,Q under RSK
both of that shape lambda, and all share the
same P. (or is it Q? one for left cells the other for right) 