Back and forth...back and forth...back and forth...are you sea-sick yet? The sway, or rocking motion, of a boat is too much for many stomachs.
People can sway if they're dizzy, tilting from side to side as they walk. On a windy day you can see trees swaying and bending in the wind. Swaying is usually a gentle motion, but if you're easily swayed, you're in trouble. That means you're easily influenced by others. People of power often "hold sway" over their followers, controlling them with the seeming ease of the breeze.