When the market makes a bullish move, this means it is moving upwards, and when it makes a bearish move, this means it is moving downwards. Bulls are usually people who have bought a certain currency pair or financial instrument (or those who expect it to go up), and bears are those who sold the instrument or expect it to go down.
The names come from the typical movements of a bull and a bear. A bull usually butts up his opponent whereas a bear smacks his victim down. In the same sense, bulls want to take the price up, and bears want to take the price down. The price eventually moves in the direction of the strongest of either one of those two sides.
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When talking about markets, a bull market is a market that is predominantly controlled by bulls, and a bearish market is a market where the bears are prevailing. One example of a bull market is the long trend up that lasted for more than ten years in the S&P500 after the financial crisis of 2008. An example of a bear market is the one that followed when stocks declined sharply because of the lockdowns in many economies due to the COVID-19 pandemic.