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The precise source of the phrases "bull market" and "bear market" are difficult to understand. The Oxford English Dictionary cites an 1891 had used of the term "bull market".
The most common etymology points to London bearskin "jobbers" (market makers who would put up for sale bearskins before the bears had actually been caught in contradiction of the proverb ne vendez pas la peau de l'ours avant de l’avoir tué ("don't put up for sale the bearskin before you've killed the bear")—an caution against over-optimism. By the time of the South Sea Bubble of 1721, the bear was also connected with short selling; jobbers would sell bearskins they did not own in anticipation of falling prices, which would enable them to buy them soon after for an additional profit. Another reasonable origin is from the word "bulla" which literally means bill, or contract. When a market is rising, holders of contracts for future delivery of a commodity see the significance of their contract increase. Nevertheless in a falling market, the counterparties—the "bearers" of the commodity to be delivered, win because they have locked in a future delivery price that is higher than the current price.
The word "bull" plays off the market's returns being "full" while "bear" alludes to the market's returns being "bare".
Some analogies that have been used as mnemonic diplomacy