Volatility is the size of price movement, without regard to direction. It is a separate dimension from the price level, it can be traded on its own, and it behaves quite differently: volatility is mean-reverting and clusters in time, while price levels are close to a random walk.
Two distinct quantities share the name and confusing them causes most of the errors in the subject. Realised volatility is a measurement of how much price actually moved over a past window. Implied volatility is extracted from option prices and describes what the market is charging for future movement. The first is a statistic about history; the second is a price.
Options are the instrument through which volatility is traded, because an option's value depends on how far the underlying might move before expiry. Buy an option and you are long volatility; sell one and you are short it, with a payoff profile that is small and positive most of the time and occasionally very negative. That asymmetry is the whole character of the business.
The market prices volatility differently by strike and by expiry, which means the object being traded is not a number but a surface. See the volatility surface for what that shape is and term structure for how it varies with time to expiry.