Need, necessity, exigency may all denote either a state or condition requiring something as essential or indispensable or the thing required.
Need implies pressure and urgency arising either from external or internal causes or forces; it may merely suggest the call of an appetite or demand for emotional or intellectual satisfaction or it may imply circumstances (as a breakdown or interruption of activity, poverty, a storm, or a threat of war) that expose a lack of or create a demand for something indispensable (as to the well-being, protection, security, success, or functioning of those or the one concerned).
Necessity, though often interchanged with need, usually carries a stronger suggestion of an imperative demand or of a compelling cause.
Necessity may also apply to a compelling principle or abstract force inherent in nature or in the constitution of a thing and inevitable in its operation or inescapable in its results.
Exigency (see also JUNCTURE ) implies the compulsion of necessity or occasionally of an inherent compelling principle, especially as a result of such special circumstances as a crisis, an emergency, or an accident, that imposes severe restrictions or great stress and strain; in either case, the term emphasizes, more than either of the preceding words, extreme urgency, demands of a peremptory and exacting character, and difficulties that cannot be easily overcome.