the order of a method

How do you put a single number on how good a numerical method is? You ask how fast its error shrinks as you refine the step. A method that halves its error when you halve the step is improving slowly; one that cuts its error sixteenfold for the same refinement is improving fast. The order is exactly this rate — the power p such that the global error behaves like h^p. Higher order means dramatically more accuracy for the same effort.

Precisely, a method has order p if its global truncation error at a fixed final time is of size h^p — that is, proportional to h^p for small h. Equivalently (and this is how order is usually computed), its local truncation error per step is of size h^(p+1); the extra power is lost because covering a fixed interval takes about 1/h steps whose errors accumulate. You find p by Taylor-expanding both the true solution and the method's update and seeing how many leading terms agree: match through the h^p term, and the first mismatch is at h^(p+1), so the order is p. Euler is order 1; improved Euler and midpoint are order 2; RK4 is order 4.

Order is the headline figure of merit, and it explains the entire hierarchy of methods. Halving h multiplies the error by (1/2)^p, so for order 4 you gain about four decimal digits for every tenfold refinement, versus one digit for order 1. That is why RK4 crushes Euler in practice. But order is not the whole story: it describes only the truncation error as h -> 0 in exact arithmetic, and says nothing about stability or round-off. A high-order method that is unstable on your problem, or pushed past the round-off floor, will not deliver its promised accuracy.