When you evade something, you escape it. You could evade a police chase by slipping into a secret alley, or you could evade your mother's questions about the missing cookies by slipping into another topic.
Other things people like to evade? Death. Taxes. Creepy ex-boyfriends. The verb evade comes from Latin roots ex ("away) and vadere ("to walk"), meaning literally "to walk away or to escape." Definitely what you want to do with creepy ex-boyfriends.