As a prefix, 非 means "non-" and attaches to nouns/adjectives: 非金属 (non-metal), 非正式 (informal, unofficial), 非法 (illegal), 非营利 (non-profit), 非暴力 (non-violent).
grammar
非 as a prefix creates a new compound word, unlike 不 which simply negates a verb or adjective. Compare 非正式 (informal, a category) with 不正式 (not formal, a judgment).
Components
pictograph
非fēi
not; non-; wrong
Pictograph of a pair of wings spread apart — two mirrored sets of feathers facing away from each other in opposite directions. From that picture of things splayed in opposite directions came the meaning 'opposite, contrary, not.' 非 is its own Kangxi radical (#175), though few modern characters file under it. The visual symmetry is the key to remembering: two wings turning away from each other = negation.
