'Companion species' is a term coined by Donna Haraway which describes the mutual, two-way relationship between human and nonhuman species (including animals like dogs but also extending to bacteria and insects). Haraway argues that companion species are co-constituted within their shared world; we make each other what we are and hence do not pre-exist our relating. Understanding dogs as companion species means understanding that our very being is fundamentally linked to that of dogs - both socially, historically and biologically. What I like about Haraway’s explication of this idea is her emphasis on the fact that dogs are not babies or toys or projections of oneself but dogs, joined with us in 'significant otherness'. 

(If you would like to read more about this do turn to ‘The Companion Species Manifesto: Dogs, People, and Significant Otherness’ by Donna Haraway)