The number 1 gets a lot more press than -1, and the concept of heterozygositygets more attention than homozygosity. Concretely the difference between the latter two is rather straightforward. In diploid organisms the genes come in duplicates. If the alleles are the same, then they're homozygous. If they're different, then they're heterozygous. Sex chromosomes can be an exception to this because in the heterogametic sex you generally have only one copy of gene as one of the chromosomes is sharply truncated. This is why in human males are subject to X-linked recessive traits at such a great frequency in comparison to females; recessive expression is irrelevant when you don't have a compensatory X chromosome to mask the malfunction of one allele. Of course recessive traits are not simply a function of sex-linked traits. Consider microcephaly, an autosomal recessive disease. To manifest the trait you need two malfunctioning copies of the gene, one from each parent. In other words, you exhibit a homozygous genotype with two mutant copies. I suspect that this particularly common context of homozygosity, recessive autosomal diseases, is one reason why it is less commonly discussed outside of specialist circles: there are whole cluster of medical and social factors which lead to homozygosity which are already the focus of attention. The genetic architecture of the trait is of less note than the etiology of the disease and the possible reasons in the family's background which might have increased the risk probability, especially inbreeding. In contrast heterozygosity is generally not so disastrous. Even if functionality is not 100%, it is close enough for "government work." The deleterious consequences of a malfunctioning allele are masked by the "wild type" good copy. The exceptions are in areas such as breeding for hybrid vigor, when heterozygote advantage may be coming to the fore. The details of complementation of two alleles matter a great deal to the bottom line, and the concept of hybrid vigor has percolated out to the general public, with the more informed being cognizant of heterozygosity.
But homozygosity is of interest beyond the unfortunate instances when it is connected to a recessive disease. Like heterozygosity, homozygosity exists in spades across our genome. My 23andMe sample comes up as 67.6% homozygous on my SNPs (which are biased toward ~500,000 base pairs which tend to have population wide variation), while Dr. Daniel MacArthur's results show him to be 68.1% homozygous across his SNPs. This is not atypical for outbred individuals. In contrast someone whose parents were first cousins can come up as ~72% homozygous. This is important:
zygosity is not telling you simply about the state of two alleles, in this case base pairs, it may also be telling you about the descent of two alleles.
