The (asymptotic) convergence orders (rates of convergence) are notions introduced for measuring the speed of convergent sequences from R, R^n, normed or metric spaces.
The asymptotic constant appears in the classical definition (the C-order), while other definitions are given by the Q-order and the R-order.
An iterative method is said that has high convergence order/speed, if the iterates converge with at least superlinear order. The most known such method is the Newton method, which, under some standard assumptions, has quadratic convergence.
The computational convergence orders (numerical convergence speed) are defined with the aid of the elements of a sequence known at a step k.
In a recent survey paper, E. Catinas has made a rigorous presentation of all the aspects concerning this topic: definitions, connections, historical aspects.