In the parlance of Wall Street, volatility means range expansion. If the S&P 500, for example, goes through a period where the average daily range of prices is say 10 points and then enters a period where the average range expands to 25 points, volatility has increased. Volatility says nothing about direction. Volatility can increase all in one direction, and a choppy market can occur with low or contracting volatility.
Let's view an example from the October correction:
Likewise, you can glance back to the minor top that formed in June/July. Prices chopped back and forth in a 50-point range about four times before sinking into August. This choppiness was associate with an ATR of 10-12 versus the spike to ATR 30 which occurred during the October correction.
The distinction is important in trading. For example, option writers love choppy markets, which creates a theta burn, allowing them to cash in on premiums. Volatile markets can also be useful for other options strategies, including naked buying (if you get the direction right) and straddles and strangles (if you don't pay too much for implied volatility).
For those that prefer a more technical explanation, the Wikipedia article on financial volatility provides a decent overview.