A martingale is a process, always associated with a filtration, or the unfolding of a story. Almost always ) the unfolding has a time element.
 except trivial cases like “revealing one poker card at a time” … don’t spend too much time on that.
In the Ito formula context, (local) martingale basically means zero dt coefficient. Easy to explain. Ito’s calculus always predicts the next increment using 1) revealed values of some random process and 2) the next random increment in a standard BM:
dX = m(X, Y, …, t) dt + S1(X, Y…, t)dB1 + S2(X, Y…, t)dB2 +…
Now, E[dX] = 0 for a (local) martingale, but we know the dB terms contribute nothing.
counter-example – P xxx [[Zhou Xinfeng]] has a simple, counter-intuitive illustration: B3 is NOT a martingale even though by symmetry E[B^3] = 0. (Local) Martingale requires
E[B^3 | last revealed B_t value] = 0 , which doesn’t hold.