In this paper, we study the sequential warped product manifolds, which are the natural generalizations of singly warped products. Many spacetime models that characterize the universe and the solutions of Einstein's field equations are known to have this new structure. For this reason, first, we investigate the geometry of sequential warped product manifold under some conditions of concircular curvature tensor. We also study the conformal and gradient almost Ricci solitons on the sequential warped product. These conditions allow us to obtain some interesting expressions for the Riemann curvature and the Ricci tensors of its base and fiber from the geometrical and the physical point of view. Then, we give two important applications of this concept in the Lorentzian settings, which are sequential generalized Robertson-Walker spacetimes and sequential standard static spacetimes and obtain the form of the warping functions. Also, by considering generalized quasi Einsteinian conditions on these spacetimes, we find some specific formulas for the Ricci tensors of the bases and fibers. Finally, we terminate this work with some examples for this structure.