We derive guaranteed, fully computable, constant-free, and sharp upper and lower a posteriori estimates on the algebraic, total, and discretization errors of finite element approximations of the Poisson equation obtained by an arbitrary iterative solver. The estimators are computed locally over patches of mesh elements around vertices and are based on suitable liftings of the total and algebraic residuals. The key ingredient is the decomposition of the algebraic error over a hierarchy of meshes, with a global solve on the coarsest mesh. Distinguishing the algebraic and discretization error components allows us to formulate safe stopping criteria ensuring that the algebraic error does not dominate the total error. We also prove equivalence of our total estimate with the total error, up to a generic polynomial-degree-independent constant. Numerical experiments illustrate sharp control of all error components and accurate prediction of their spatial distribution in several test problems. These include smooth and singular solutions, higher-order conforming finite elements, and different multigrid methods as well as the preconditioned conjugate gradient method as the iterative solver.