In this paper, we focus on a class of convex optimization problems subject to equality or inequality constraints and have developed an Accelerated Inexact Augmented Lagrangian Method (AI-ALM). Different relative error criteria are designed to solve the subproblem of AI-ALM inexactly, and the popular used relaxation step is exploited to accelerate the convergence. By a unified variational analysis, we establish the global convergence of AI-ALM and its sublinear convergence rate in terms of the primal iterative residual, the objective function value gap and constraint violation, respectively. Numerical experiments on testing the image restoration problem with different types of images indicate that AI-ALM is effective and promising. In appendix, we also extend AI-ALM to solve a general multi-block problem and briefly discuss convergence of the extended method.