What specific biochemical processes are involved in inducing meiosis rather than mitosis? Why are gonadal cells the only cells in the human body which do undergo meiosis?
It's a difficult question because every step in the development of a germ cell is ultimately necessary for the final differentiation, which includes a meiotic division. Meiosis requires a lot of specialized components to pair and segregate homologues, to induce and resolve recombination, etc. What starts it all is still largely unknown. There are plenty of mutants that halt the process, but these are required along the way, so damaging the pathway ultimately stops it from progressing. At least one study has been able to initiate the program of meiosis in yeast:
Induction of meiosis in Saccharomyces cerevisiae depends on conversion of the transcriptional represssor Ume6 to a positive regulator by its regulated association with the transcriptional activator Ime1. I Rubin-Bejerano, S Mandel, K Robzyk, and Y Kassir
Basically, they turned on a transcription factor, which activated an entire suite of downstream genes necessary for meiosis. In essence, they turned on the "meiosis pathway." Bear in mind this is yeast, so does't have separate germ cells, but the concept is probably the same.
All other cells are diploid. Only in germ cells does the organism induce reductional divisions (to make haploid gametes for ultimate fusion in the zygote of the next generation). Creation of haploid somatic cells would uncover recessive lethal mutations and cells would die. In sperm and eggs, which do not express any genes until after fertilization and karyogamy, this is not a problem.
Ultimately, it seems that follicle-stimulating hormone (FSH), released from the pituitary gland, may be the direct signal for spermatocytes to enter meiosis . It's worth noting, though, that these results come from in vitro studies, where spermatocytes were co-cultured with all cell types of seminiferous tubules in an artificial medium, which makes it a rather crude approximation of in vivo conditions: mimicking the complete spermatogenic cycle in vitro remains to be done .
It's a little bit more complicated with oocytes. They start meiosis during prenatal development, are arrested in diplotene stage of the first meiotic division, and then a surge of luteinizing hormone (LH), just before ovulation, makes them resume. But they are arrested again in the last phase, and only after conception do they finish meiosis. The pause occuring in prenatal development is thought to be induced by some kind of an "arrester" secreted by cells in the oocyte environment (specifically, mural granulosa cells) and sustained by high cAMP levels, that are produced in response to both intrinsic and extrinsic stimuli. 
Another matter, and quite interesting, is the issue of timing: meiosis starts in a distinct moment of a male mammal develompent, and then goes on continuously, but mammal oogenesis is a series of starts and stops. More on this can be found in  and .
 Tesarik, J., Guido, M., Mendoza, C. & Greco, E. Human spermatogenesis in vitro: respective effects of follicle-stimulating hormone and testosterone on meiosis, spermiogenesis, and Sertoli cell apoptosis. J. Clin. Endocrinol. Metab. 83, 4467–4473 (1998). PMID: 9851795. Free access.
 Sousa, M., Cremades, N., Alves, C., Silva, J. & Barros, A. Developmental potential of human spermatogenic cells co-cultured with Sertoli cells. Hum. Reprod. 17, 161–172 (2002). PMID: 11756382. Free access.
 Zhang, M. & Xia, G. Hormonal control of mammalian oocyte meiosis at diplotene stage. Cell. Mol. Life Sci. Epub ahead of print. (2011). PMID: 22045555.
 Albertini, D.F. & Carabatsos, M.J. Comparative aspects of meiotic cell cycle control in mammals. J. Mol. Med. 76, 795–799 (1998). PMID: 9846949.
Let's start by answering the second part of your question first. I can ask you a similar question like "Why do parietal cells secrete HCl?" or "Why are skeletal muscle cells multinucleated?". The reason why the cells in our body carry out different functions and undergo difference processes, despite having the same DNA, can be due to a multitude of factors. Here are some off the top of my head:
Now to answer what the specific mechanism in which gonadal cells undergo meiosis:
For one, histone modification (and therefore differential gene expression) has been implicated in the regulation of spermatogenesis. You can read about it in this paper: Song N, Liu J, An S, Nishino T, Hishikawa Y, Koji T. 2011. Immunohistochemical Analysis of Histone H3 Modifications in Germ Cells during Mouse Spermatogenesis. Acta histochemica et cytochemica 44: 183–90.
At first glance, I didn't find anything on oogenesis; will update if I see anything.