Volatility is a measure of the variance, deviation, or movement of a stock.
Volatility is all the extra movement of a stock or other security over and above (and below) a line of averages. Put another way, it is a measure of how many changes in price, and by how much, a security experiences over an amount of time.
Computations of Standard Deviation and Variance are measures of the degree of volatility which exists in the movement of a stock. Volatility will also be measured relative to a benchmark index, and the degree to which a security adheres or deviates from the benchmark is called Beta. People will also trade on derivatives of the VIX, which is the volatility index of the S&P 500.
Increased volatility brings increased uncertainty, and times of volatility often cause investors to seek stability, but more active/speculative investors will be excited by the volatility, because price changes are where money is made, both by gains in long or short positions and derivatives, but also, in the case of advisors and brokers, by commissions and billable time. Volatility is not necessarily synonymous with risk, but it is a strongly correlated concept.
In options trading, implied volatility is a term used for the marketability and potential profitability that is priced into an option in which the underlying is experiencing volatility, and it represents a sentiment measure of the volatility expected going forward, and, of course, sentiment will often affect or even determine the actual prices and volatility of securities.