A bull market is an indication that investors/traders expects share prices to rise and have optimistic view of the market trend.
An investor/trader who expects share prices to fall is called a bearish trader. The conditions of prolonged drop in markets is called a bear market. Bear market is an indication that buyers are not present and prices begin to fall.
However, why certain animal names are used to indicate the market trend is unclear. Possibly, the terms are derived from the animal behavior on how it attacks its opponent.
For eg: A bull pushes its horns up into the air while a bear swipes its paws down. These both actions are compared to movement of the market.