The article Alternating-Sign Matrices and Domino Tilings by Noam Elkies, Greg Kuperberg, Michael Larsen, and myself got the whole Aztec diamond game started. You should read sections 1, 2, 3, 6, 7, and 8. The brave can read section 4, and the super-brave can read section 5. (Unfortunately, all the illustrations appear at the end. Even worse, Figures 6-11 don't show up for all viewers; for other viewers, only Figure 7 is missing. If you have trouble getting all the figures, ask me to send you a copy of the article, or else look up the article in the Journal of Algebraic Combinatorics, volume 1, pages 111-132 and 219-234 (1992).)
Eric Kuo's new proof of the formula for counting domino tilings of the Aztec diamond is even simpler than the four proofs found by Elkies, Kuperberg, Larsen, and Propp. (Furthermore, Eric's method can be vastly generalized.)
Jim Propp's "urban renewal" approach to studying perfect matchings (and weighted perfect matchings) of the graphs dual to Aztec diamonds. Like Kuo's method, this one is extremely general; I suspect that at a deep level they are equivalent.
Chris Douglas' description of how he applied urban renewal to the "every-second-diagonal" problem is a wonderful illustration of the steps involved in doing original research.
My unpublished tech report Dimers and Dominoes shows that, even though the formula for counting domino tilings of squares and rectangles is nowhere near as nice as the formula for counting domino tilings of Aztec diamonds, there's still some sweet mathematics there, and a nice application of basic Fourier theory.
My unpublished article Twenty Open Problems in Enumeration of Matchings gives a sense of the wide variety of open questions that arise when one starts playing around with perfect matchings of planar graphs. But beware! Many of these problems have already been solved; to find out the latest on which of them have been settled, see the progress report that accompanies it.
Finally, you should read the research proposal that I submitted to the NSF and NSF when I applied for the grant that is funding my use of undergraduate research assistants. This gives a sense of my broader goals, which go beyond merely generating and proving new combinatorial identities. I want to understand how boundary conditions in multi-dimensional stochastic systems can have an impact far away from the boundary, in the interior. Ultimately, I hope that the ideas that arise in an attempt to understand these systems rigorously (such as the idea of "effective fields") will have applicability to other sorts of non-homogeneous systems. Twenty Open Problems in Enumeration of Matchings
If you have any questions, don't hesitate to contact me (Jim Propp).