local truncation error
Suppose you cheat: you start a single step from a point that is exactly on the true solution curve. You take one perfect-data step with your method, and you ask — how far off is my landing point from where the true solution actually is? That gap, the error made in just one step starting from exact values, is the local truncation error. It is the method's honesty checked under the kindest possible conditions: no accumulated past mistakes, just the error of the formula itself for a single hop.
Precisely, it measures how badly the method's update formula fails to match the true behaviour of the solution over one step h. The name 'truncation' is the giveaway: a method is really a Taylor expansion of the true solution with the higher-order terms chopped off, and the local truncation error is exactly the size of the terms thrown away. For Euler's method the discarded terms start at (h^2 / 2) * y'', so the local truncation error is of size h^2 — that is, proportional to h^2. A method whose local error is of size h^(p+1) is said to have order p.
Local truncation error matters because it is the seed of all the trouble. It is the cleanest thing to compute — just compare Taylor expansions — and it directly tells you a method's order. But it is local, measured per step under ideal conditions; the error you actually see at the end of a long run is the global truncation error, which is roughly the local error times the number of steps. The usual rule of thumb: if the local error is of size h^(p+1), the global error is of size h^p — one power smaller, because the errors accumulate over about 1/h steps.
Euler's method comes from y(t+h) = y(t) + h y'(t) + (h^2/2) y''(t) + ... The method keeps the first two terms and drops the rest. The first dropped term, (h^2/2) y'', is the local truncation error — of size h^2 — which is why Euler is a first-order method (global error of size h).
The local error is the Taylor terms the method throws away in a single step.
Do not confuse local with global error. The local truncation error is one power of h smaller-looking than the method's order, because order refers to the GLOBAL error after about 1/h steps accumulate. A method of order p has local error of size h^(p+1).