Algebraic MultiGrid (AMG) method, is one of the most efficient numerical techniques to solve large-scale linear system, especially for those arising from discretization of systems of partial differential equations (PDEs). In this work, we try to understand how and why an algebraic multigrid method in a more abstract level. We develop a unified framework and theory that can be used to derive and analyze different algebraic multigrid methods in a coherent manner. Given a smoother $R$ for a matrix $A$, such as Gauss-Seidel or Jacobi, it is well-known that the optimal coarse space of dimension $n_c$ is the span of the eigen-vectors corresponding to the first $n_c$ eigen-values of $\bar RA$ (with $\bar R=R+R^T-R^TAR$). We prove that this optimal coarse space can be obtained by a constrained trace-minimization problem for a matrix associated with $\bar RA$ and demonstrate that coarse spaces of most of existing AMG methods can be viewed some approximate solution of this trace-minimization problem. Furthermore, we develop a general approach to the construction of a quasi-optimal coarse space and we prove that under appropriate assumptions the resulting two-level AMG method for the underlying linear system converges uniformly with respect to the size of the problem, the coefficient variation, and the anisotropy. Our theory applies to most existing multigrid methods, including the standard geometric multigrid method, the classic AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG, and spectral AMGe. Finally, we apply AMG methods to two application problems, one is on electrokinetics flow and the other is on reservoir simulation, which provide numerical results to demonstrate the efficiency and robustness of AMG methods.