There are many definitions of species - look at the wikipedia page for an overview (this is a large and tricky subject). Some hybridization is not uncommon between "proper" species (often in narrow contact zones), also with fully or partially fertile hybrids. Definitions of species in these cases often comes down to genetic similarity/dissimilarity of populations. Even though hybridization is taking place it usually does so at very low rates, so the gene pools stay relatively distinct. The biological species concept (which is what you are referring to) is also problematic when it comes to e.g. ring species.
As for wild cats and domestic cats, my feeling is that the majority of the research community see them as separate species (as does IUCN and Mammal Species of the World ), but some consider them to be the same species (Driscoll et al. 2009), with the domesticated cat as Felis silvestris catus. In the study by Driscoll et al. (2009), domesticated cats fall in the same mtDNA clade as F. silvestris lybica.
There is also evidence that hybridization can help adaptation by facilitating gene transfer (even though hybrid fertility/viability might be low), so you might want to check out e.g. results from the Heliconius Genome Consortium (2012).