necessary and sufficient
Think of a key and a door. A condition is sufficient if having it guarantees the door opens; it is necessary if the door cannot open without it. When a single condition is both — the door opens exactly when you have it — the condition and the outcome are interchangeable.
A condition C is sufficient for a conclusion P when C implies P; it is necessary for P when P implies C; it is necessary and sufficient when both implications hold, written C if and only if P. Such a biconditional says C and P are logically equivalent: each is true precisely when the other is.
Necessary-and-sufficient conditions are the gold standard of an analysis theorem, because they replace a vague property by an equivalent, checkable one. The Cauchy criterion, for instance, gives a condition equivalent to convergence that never mentions the limit. The common error is to confuse the two directions: ‘differentiable implies continuous’ is true, but its converse is false (|x| is continuous yet not differentiable at 0), so continuity is necessary but not sufficient for differentiability.
An integer n is even if and only if n^2 is even. Forward: if n is even then n^2 is even. Backward: if n^2 is even then n is even (its contrapositive: odd n gives odd n^2). Both directions hold, so the conditions are interchangeable.
‘If and only if’ demands both directions be proved, not just one.