zero-stability
A multistep method builds each new value from several previous ones, and this recurrence has a hidden danger that one-step methods do not. The formula may have its own internal dynamics — spurious solutions that have nothing to do with your differential equation, born purely from the recursion. If those parasitic modes grow rather than stay bounded, they will swamp the genuine answer, no matter how accurate each step looked. Zero-stability is the condition that keeps those parasitic modes from exploding.
Precisely, zero-stability examines the simplest possible test, the trivial equation y' = 0, in the limit of zero step size. The method then reduces to a pure linear recurrence among the y values, and its behaviour is governed by the roots of a characteristic polynomial built from the method's coefficients. Zero-stability is exactly the root condition: every root of that polynomial must have magnitude at most 1, and any root with magnitude exactly 1 must be simple (not repeated). Pass this, and small starting errors stay bounded as you take more steps; fail it, and they grow without limit.
Zero-stability is the crucial second ingredient of convergence. Consistency says the method aims at the right equation; zero-stability says the method does not amplify the errors it makes along the way; together they give convergence (the Dahlquist equivalence theorem). It also explains why you cannot make a multistep method arbitrarily high order for free: the Dahlquist barrier says a zero-stable linear multistep method of a given number of steps can only reach a limited order, so chasing higher order eventually costs you stability. Note this is a different stability from absolute stability, which concerns a fixed nonzero h, not the h -> 0 limit.
Apply a multistep method to y' = 0: it becomes a recurrence like y_(n+1) = (coefficients)(y_n, y_(n-1), ...). Its characteristic polynomial's roots must all lie in or on the unit circle, with any boundary root simple. The Adams families satisfy this; certain higher-order 'optimal' formulas violate it and are useless despite their order.
The root condition: spurious recurrence modes must not grow.
Zero-stability (the h -> 0 limit, governs convergence) is NOT the same as absolute stability (a fixed nonzero h, governs behaviour on a given problem). A method can be zero-stable yet still blow up at too large a step size.