How do you determine where a coin was minted?

If you're referring solely to US Coins, it requires a knowledge of mint mark letters and their positions. If you have a specific coin or coins in mind, one good site is linked below. It shows the positions and letters for every circulating US coin ever minted.

Unfortunately there's no general answer that applies to all coins because positions vary by date and denomination. Since 1968 the position has been pretty much standardized near the date, and since 1980 ('79 for dollars) all Philadelphia coins except cents have a P mint mark.

Before those years it was pretty much a free-for-all. Up till 1968 most but not all US coins had mint marks on the reverse side but the position could be anywhere within the design - under the eagle on some higher-denomination coins, near the edge on others, or in some other place depending on what might fit conveniently within the design. To make things even more confusing, nearly all coins minted in Philadelphia up till 1980 didn't have any mint marks at all!