The story most often told relates to how each animal is said to attack. A bull will thrust its horns into the air, while a bear will swipe down. These actions metaphorically reflect the movement of a market, with bull markets trending up and bear markets trending down.
A bullish market is a time when the demand is higher than the supply of shares and results in the rising of the share prices. A bearish market is a time when the supply is higher than the demand for the shares and results in the fall of the prices of the shares.