The opposite of a bull market is a bear market where the market is trending lower.
In common usage a bull market in the post-war period has probably lasted at least 2 years and bear markets about the same. Like recessions they are often best recognised after the fact. Dow Theory attempts a description of the character of these market movements; there are also many studies of the history of the markets e.g the book below by Hurst.
Both bull and bear markets may be fueled by sound economic considerations and/or by speculation. An exaggerated bull market fueled by over-confidence and/or speculation can lead to a stock market bubble. At the other extreme an exaggerated bear market, that tends to be associated with falling investor confidence, can lead to a stock market collapse and a capitulation phase (abandonment of hope).
Expectations play a large part in financial markets and in the changes from bull to bear environments. More precisely, attention should be paid to positive surprises and negative surprises. The tendency is for positive surprises to characterise a bull market (when the news continually tends to exceed investor's expectations) and conversely negative surprises tend to characterise the bear market (with expectations disappointed).