The name originates from the bull's manner of striking its horns upward. The opposite is a bear market, a period of prolonged decline. There is no universally accepted formal criterion; a conditional threshold of index change of roughly one-fifth from the previous extreme is often used.
Bulls are participants betting on price increases, bears are those expecting declines. These terms describe position, not character: the same investor can be a bull on one security and a bear on another.
The practical value of this concept is limited. Market phases can only be reliably identified in hindsight, and attempting to predict a reversal statistically underperforms regular scheduled investments. A more useful implication is understanding that prolonged growth increases risk tolerance and encourages buying at peaks, so portfolio allocation decisions are better made according to predetermined rules rather than market sentiment.