I can provide something of an answer for China, which will probably be useful for situating other answers given that between a third and half of all the silver mined in the New World ended up in Chinese markets.
Before Spanish silver mines in modern-day Bolivia and Peru started producing large quantities of bullion and shipping it over the Pacific, the predominant currency for everyday transactions in Ming Chinese markets was copper coinage. Early in the dynasty, however, the state mandated that taxes be paid in-kind (i.e. the actual goods, food, or products you produced, like millet, rice, or raw silk). Tax obligations were quite rigid, however, and in many cases local/regional governments mandated in-kind taxes that made little economic sense for their subjects to provide. If you were the head of a household with more resources available than the average peasant family, and the main thing you produced was raw silk, but your in-kind tax obligation was rice, you were in something of a conundrum. Why produce rice when silk (or citrus fruits or some other product) made more economic sense for you? So the solution was naturally to sell your raw silk and use some of the profit to buy rice, and pay your in-kind tax with that rice. Long story short, this fueled the rise of petty commercial markets - systems that had existed in abundance during the Song but declined somewhat in the Yuan.
Over time, specific regions of China started to specialize in particular products. If you were a wealthy Confucian scholar-official living in late-fifteenth or sixteenth-century China, and you wanted to buy embroidered items, you'd best buy them from Hangzhou, or else your wealthy Confucian scholar-official friends would know you were a miser who owned second-rate goods. If you wanted the finest silks, you bought them from Suzhou. If you wanted the finest paper and porcelain, you bought it from Jiangxi. If you lived in Huian and wanted the best lychees or longans, Heaven forbid you buy them from local suppliers, which only produced mediocre lychees and longans; everyone knew the best came from up the coast in Fengtian and down the coast in Quanzhou.
But your question was about currency, so where am I going with this description of markets? As luxury markets grew dramatically in Ming China, merchants rose in social and economic standing. Merchants needed convenient methods to make transactions when dealing with large quantities of luxury goods. A fruit merchant from Quanzhou shipping a boatload of lychees up the coast to his partner redistributors could accept payment in copper coins, but you'd need a lot of copper coins to pay for large shipments. Silver, however, is more valuable and makes large-scale, expensive transactions more manageable. So, in short, the influx of silver into China filled an economic gap and helped fuel large-scale market forces, while copper coins remained the ideal currency for small-scale petty commerce, especially in parts of inland China where silver might be less abundant (since it arrived in China from the New World via the coast), or in places where the value of silver was out of sync with its value nearer to the coast, making it a less attractive form of economic lubrication.
Why was all this silver coming to China in the first place? There are doubtless other posts on this subreddit that can provide a better answer than I, but the long and short of it was: Europeans wanted Chinese goods, but Chinese markets didn't want European goods. So Europeans paid for Chinese goods with the only thing they had that Chinese merchants and markets desired: silver.
Sources:
Timothy Brook, The Confusions of Pleasure: Commerce and Culture in Ming China
William Atwell, "Ming China and the Emerging World Economy," in The Cambridge History of China, Volume 8