Single-step vs. multistep ODE methods
Euler and Runge-Kutta's fresh start every step vs. Adams-Bashforth-Moulton's memory of the points already computed.
A single-step method only ever needs the CURRENT point to advance -- simple to start (including after an adaptive step-size change) and the natural default. A multistep method reuses several already- computed past points instead of sampling several new slopes per step, which is cheaper PER STEP once it's running -- but it needs a single-step method to bootstrap its first few points, and switching step size mid-run is more disruptive. Reach for RK4 by default; switch to an adaptive single-step method (Runge-Kutta-Fehlberg) when the solution's behavior changes pace across the interval, and to a multistep predictor-corrector pair when function evaluations are themselves expensive and you're taking many steps at a fixed size. Either family can go unstable on a STIFF equation without a small enough step -- that's a separate concern from which family you picked.
Single-step methods -- only need the last point
Sample several slopes within one step and combine them -- the general family Euler's method is the crudest member of.
RK2 -- one slope at the start, one at the estimated midpoint.
For \dfrac{dy}{dx}=f(x,y) with step size h, RK4 advances from (x_n, y_n) to (x_{n+1}, y_{n+1}) by sampling the slope at four points and taking a weighted average: k_1 = f(x_n, y_n), k_2 = f\!\left(x_n + \tfrac{h}{2},\, y_n + \tfrac{h}{2}k_1\right), k_3 = f\!\left(x_n + \tfrac{h}{2},\, y_n + \tfrac{h}{2}k_2\right), k_4 = f(x_n + h,\, y_n + h\,k_3), y_{n+1} = y_n + \frac{h}{6}\left(k_1 + 2k_2 + 2k_3 + k_4\right).
Show proof
k_1 is the slope at the left endpoint; k_2 and k_3 are two independent estimates of the slope at the midpoint, the second refining the first by using k_2's own predicted y-value instead of k_1's; k_4 is the slope at the right endpoint, predicted using the midpoint slope k_3. Weighting these 1:2:2:1 and dividing by 6 reproduces Simpson's rule's own weighting for approximating an integral by samples at both endpoints and (doubly) the midpoint. Carrying this weighted average through a Taylor expansion of the true solution shows it matches every term through h^4, giving a local truncation error of O(h^5) per step and a global error of O(h^4) — two full orders better than Euler's method, at the cost of four function evaluations per step instead of one.
Adjusts h on the fly by comparing two embedded estimates of different order.
Multistep methods -- reuse several past points
Uses y and f from SEVERAL previous points, not just the current one, to advance.
An explicit (Adams-Bashforth) guess, refined by an implicit (Adams-Moulton) correction.
Stability considerations for either family
The region of h\lambda where a method's own error stays bounded rather than blowing up.
Forces an impractically small h on an explicit method of EITHER family, regardless of accuracy needs.