Modern numerical techniques constitute the foundation of computational applied mathematics, enabling the solution of complex mathematical models governing engineering systems, physical phenomena, and data-driven applications. This comprehensive review examines contemporary numerical methods with emphasis on stability analysis, convergence theory, error control mechanisms, and computational efficiency considerations essential for reliable scientific computation. The study investigates classical approaches including finite difference and finite element methods alongside advanced techniques such as spectral methods, mesh-free formulations, and emerging hybrid computational frameworks. Particular attention is devoted to mathematical foundations of stability theory, including Courant-Friedrichs-Lewy conditions, von Neumann analysis, and energy stability methods, as well as convergence criteria and error estimation strategies that ensure accurate numerical approximations. The manuscript explores practical implementation considerations including computational cost, parallel scalability, and adaptive refinement strategies that balance accuracy against computational resources. Application domains encompass computational fluid dynamics, structural mechanics, heat transfer analysis, electromagnetic simulations, and modern data-driven modeling paradigms integrating machine learning with traditional numerical approaches. The analysis reveals that successful numerical computation requires synergistic integration of mathematical rigor, algorithmic efficiency, and domain-specific knowledge to address increasingly complex multiphysics and multiscale problems. This work provides researchers and practitioners with a systematic framework for selecting, implementing, and validating numerical methods appropriate for specific application requirements, while identifying critical challenges and promising research directions in computational applied mathematics.