The purpose of this paper is to prove some fixed point theorems for ordered contractions in partially ordered $b$-metric spaces. We consider several common fixed point and coincidence point results for four mappings in such spaces. The results obtained in this paper significantly extend numerous results in the existing literature. Further, we present some supportive examples to emphasize the potential value of our results. In addition, as applications, we verify the existence and uniqueness of solutions to a large number of equations.