Here's a very basic explanation, which would have applied to many ancient civilizations:

The minting authority would decide on the type of image that would be cast onto their coins, and hire an engraver/artist to design the image. They would engrave dies with the obverse (usually the 'head' of the coin) and the reverse (usually the 'tails' of the coin).

Then when it was time for production they heated up the metal that was going to be used (iron, bronze, gold etc), and cast it into small shapes called a planchet or flan, of the appropriate weight. It is possible that these flans were later filed down to the correct size and shape using various surfacing tools.

This flan was then held in place over an anvil, which had one of the dies embedded within it (usually the obverse die). An item called a trussel contained the other side of the die, and this was held over top of the flan and then hit on the top with a hammer to punch the image into both sides of the coin.

Often after a certain number of coins had been produced, the die would break and they would have to engrave new dies to continue the production. There are also examples where a die has become very worn or cracked but was still used to produce coinage.