From Investopedia:

BEAR MARKET

A market condition in which the prices of securities are falling, and widespread pessimism causes the negative sentiment to be self-sustaining. As investors anticipate losses in a bear market, selling continues, which then creates further pessimism. Although figures can vary, for many 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 an entry into a bear market..

BULL MARKET

A financial market of a group of securities in which prices are rising or are expected to rise. The term "bull market" is most often used to refer to the stock market, but can be applied to anything that is traded, such as bonds, currencies and commodities.

Bull markets are characterized by optimism, investor confidence and expectations that strong results will continue. It’s difficult to predict consistently when the trends in the market will change. Part of the difficulty is that psychological effects and speculation may sometimes play a large role in the markets.

The use of "bull" and "bear" to describe markets comes from the way the animals attack their opponents. A bull thrusts its horns up into the air while a bear swipes its paws down. These actions are metaphors for the movement of a market. If the trend is up, it’s a bull market. If the trend is down, it’s a bear market.

The criteria used for making a claim of a bull or bear market varies greatly between technical analysts and fundamental analysts. Even within each side their are differing opinions of what the guidelines are. One thing that has never changed throughout time is the battle between the two types of analysts.

Back in November 2007 we said that we were heading into a bear market. For us, it was mostly based on a technical analysis methodology but we were also using fundamental analysis of economic trends to add clarity to our view. Economic conditions, when viewed as trends showed us (and still do) that the fundamentals of the economy would deteriorate further and the charts were beginning to give us the warning signs that a long bear market would be upon us.

In the stock market nothing goes in one direction in a straight line. In a bear market there will always be periods where the declines stop and a rise begins. For each respite in the down ward selling the talk always turns to "is this it?, "Is this the bottom?", or "time to load the boat". In a bear market whenever there is an upward price rally we call it a "bear market rally". Meaning it is a rally within a bear market. The most important thing for traders and investors is to never fight the primary trend. As Jesse Livermore often said "Don’t fight the Trend".