To be unstable is to lack stability, meaning things could change without warning, like an unstable bookshelf that is likely to fall down. If you are unstable emotionally, you might be ecstatically happy one minute and horribly depressed and angry the next.
The adjective unstable means the opposite of stable. Stable is derived from the Latin base word stare meaning “to stand.” If a building, family, person, bridge, or chemical solution is stable it will “stand still,” even if something unexpected happens. On the other hand, if a person or thing is unstable and stress is applied, he, she, or it is likely to shift position, possibly in a dangerous way.