In this paper, we introduce the concepts of weakly and partially weakly $\alpha$-admissible pair of mappings and obtain certain coincidence and fixed point theorems for classes of weakly $\alpha$-admissible contractive mappings in a $b$-metric space. As an application, we derive some new coincidence and common fixed point results in a $b$-metric space endowed with a binary relation or a graph. Moreover, an example is provided here to illustrate the usability of the obtained results.