To do something evasively is to do it in a round about, indirect, or slightly sneaky way. When a politician responds to a question evasively, she avoids a straightforward, honest answer.
When you speak evasively, you come very close to lying — you're not being forthright, but are instead skirting around what's true. A criminal in court might answer a lawyer's questions evasively, and someone who's using a fake name might behave evasively when asked about his history. If you move evasively, you avoid being caught, and the Latin root, evadere, means "to get away or escape."