Evasive means dodging instead of answering. The minister who speaks for a full minute and tells you nothing; the friend who changes the subject every time money comes up.
You are evasive about something, you give an evasive answer, you reply evasively. The word is an accusation: it says the dodging is deliberate, which is why it stings more than shy or vague. The family runs verb, noun, adjective — evade, evasion, evasive — and tax evasion is the criminal member of it, illegal in a way that tax avoidance is not. There is a second, physical sense with no blame in it at all: evasive action is a sharp movement made to avoid a crash or an attack, so a pilot takes evasive action and so does a driver on ice.
Evasive came into English from French évasif, and standing behind that is the Latin verb ēvādō, the same verb that gave English evade.
Latin
ēvādō
French
évasif
English, today
evasive
From French évasif, from Latin ēvādō.
Etymology from Wiktionary, CC BY-SA