A stock market "bear" is an investor who believes that a particular stock or market, or the market as a whole, is headed downward. Bears often try to profit from a decline in prices. The term can be applied to many markets, including commodities/futures and bonds.
An investor who believes a particular stock or market will trend downwards is referred to as "bearish" on that stock/market.
The opposite is a "bull", who believes that the market or a particular security will go up. In the long term, the stock market has had a bullish trend.
A "bear market" is one in which the prices of securities are falling, and widespread pessimism causes the negative sentiment to be self-sustaining. A downturn of 20% or more in multiple broad market indexes, such as the Dow Jones Industrial Average (DJIA) or Standard & Poor's 500 Index (S&P 500), over at least a two-month period, is considered by most to be an entry into a bear market.
A bear market is not the same as a correction, which is a short-term trend lasting less than two months.