To start you have to know the concept of this word in the biological context, so, homology is the existence of shared ancestry between a pair of structures, or genes, in different species. But around the application of this concept exists a large discussion. Moreover, (Brigandt, 2003) propose the idea of ‘radiation of a concept’, in which explain that homology can be different depending of the biological field in where is used. For example, he exposes four fields, molecular, developmental, evolutionary and comparative biology. First, in the last ones, the goals are the comparison and even taxonomy of species and characters and the explanation of descent with modification. In phylogenetic systematics, homology is a diagnostic for monophyletic groups, which helps to characterize taxa. The role of molecular homology is the inference of information about the molecular behavior of genes and proteins. Finally, (Brigandt, 2003) concludes that the goal in evolutionary developmental biology is to figure out how and why certain structures emerge in different ontogenies. If you want a detailed review of the most famous concepts of homology you can go to (Patterson, 1982).
Here, we are going to focus in an evolutionary and phylogenetic field in which you can found a definition from homology as synapomorphy (Patterson, 1982) and the concept that reveals ancestry and descent with modification. Nevertheless, (dePinna, 1991) cited Wagner 1989 and Roth 1988, who disagree with the idea of homology as synapomorphy of (Patterson, 1982), argued that is a broader and extensive concept that should not be restricted in such way. On the other hand, (dePinna, 1991) also quote to deQueiroz 1985, he argued that homology is a notion that applies to instantaneous morphologies, while synapomorphy (according to his definition) applies to ontogenetic transformations that characterize monophyletic groups (his view of characters).
About the relation of homology and sinapomorphy, that does not have to be only attributable to Patterson 1982, (Eldredge & Cracraft, 1980) said that “…. homology can be conceptualized simply as synapomorphy (including symplesiomorphy [...])”. However respect to this affirmation is obvious noted that symplesiomorphic similarities are homologous; it is a synapomorphy at a higher level. So, for example you can found two types of homology in that sense.
Taxic homology implies a hierarchy of groups, hypotheses of monophyly, and constitutes a statement about generality of characters, or "sameness" of attributes. And transformational homology is concerned with the transformation of one structure into another.
But how recognize the existence of a homology; some authors said that there are two classes of homology, primary and secondary, the first for (Agnarsson & Coddington, 2008) is a kind of background assumption or prior knowledge that you can based in similarity, and for (dePinna, 1991)is the (theory-laden but untested) supposition that two parts are the same by inheritance. In this way, (Agnarsson & Coddington, 2008) express “In systematic biology homology hypotheses are typically based on points of similarity and tested using congruence. Indeed testing similarity is the only way to test the homology of characters, as congruence only tests their states”. However, you can say that the homology is inherent to the character and not to their states.
So, I think that if the homology hypotheses pass the test of congruence and it fit to the topology then they are right.
(Rieppel & Kearney, 2002) reviewed the topic of primary homology and argued that things can be similar to greater or lesser degrees, at different scales, and in different independent ways. The secondary homology have resisted the test of congruence, emerged as synapomorphies on a cladogram (dePinna, 1991).
The problem of primary homology is not confined to morphological data. Aligning base positions in “homologous” sequences across different taxa is a crucial phase in molecular analysis, but one method in particular, “Direct Optimization”(Wheeler, 1996), search optimal trees via direct optimization, hence varying dynamically the primary homology hypotheses. From the three tests of homology proposed to date (similarity, conjunction and congruence) only congruence serves as a test in the strict sense (dePinna, 1991), in addition is complicated establish homologies based only in similarity and in prior knowledge of a single character, in contrast, you have to integrate some characters and parameters that help to identify true homologies. However, conjunction is unquestionably an indicator of non-homology, but it does not say when a homology appears.
On the side of molecular homology there are three disjoint subtypes of homology.
Orthology: is that relationship where sequence divergence follows speciation. In (Fitch, 2000).
The true phylogeny obtained from this sequences is exactly the same as the true phylogeny of the organisms from which the sequences were obtained. Only orthologous sequences have this property.
Paralogy is defined as that condition where sequence divergence follows gene duplication. In (Fitch, 2000).
Mixing paralogous with orthologous sequences it is posible obtain a tree that has the correct phylogeny for the sequences but not for the taxa from which they derive; a gene tree is not necessarily a species tree.
Xenology is defined as that condition (horizontal transfer) where the history of the gene involves an interspecies transfer of genetic material. In (Fitch, 2000).
Agnarsson, I., & Coddington, J. A. (2008). Quantitative tests of primary homology. Cladistics, 24(1), 51–61. doi:10.1111/j.1096-0031.2007.00168.x
Brigandt, I. (2003). Homology in comparative, molecular, and evolutionary developmental biology: the radiation of a concept. Journal of Experimental Zoology. Part B, Molecular and Developmental Evolution, 299(1), 9–17. doi:10.1002/jez.b.36
Eldredge, N., & Cracraft, J. (1980). Phylogenetic Patterns and the Evolutionary Process (method and theory in comparative biology). Retrieved from http://ciib.unam.mx:8080/xmlui/handle/123456789/6819
Fitch, W. M. (2000). Homology. Trends in Genetics, 16(5), 227–231. doi:10.1016/S0168-9525(00)02005-9
Patterson, C. (1982). Morphological characters and homology. In Problems of Phylogenetic Reconstruction (pp. 21–74).
Pinna, M. C. C. (1991). CONCEPTS AND TESTS OF HOMOLOGY IN THE CLADISTIC PARADIGM. Cladistics, 7(4), 367–394. doi:10.1111/j.1096-0031.1991.tb00045.x
Rieppel, O., & Kearney, M. (2002). Similarity. Biological Journal of the Linnean Society. doi:10.1046/j.1095-8312.2002.00006.x
Wheeler, W. C. (1996). Optimization Alignment: the End of Multiple Sequence Alignment in Phylogenetics? Cladistics, 12, 1–9. doi:10.1006/clad.1996.0001