Which part of the medieval period? There's a massive chunk of history called 'the medieval period' which ranges from what was essentially (for most people) a barter economy to the more organised economies of the period just before the Renaissance...
I'm sure you'll get Medievalist in here in a bit with an opinion that is more expert than mine but here are some thoughts...
Remember, this is your world that you are making up so you can pick and choose how your own economy works. These are just suggestions based on what I have picked up on history.
Usually, economy was controlled by the monarch. They ran the royal mint and therefore made the coins that everyone used. They also set taxes (but only in wartime for most of history and, in England at least, only with the approval of parliament). Value of the currency was largely based on the value of the metal used to make it which is why, until they started to debase the currency with cheaper metals and the value was based more on the number printed on it, you could cut a coin in half or quarters to give change.
For the many people, actual coins were rare and trade was often done by barter for goods or services in exchange. Certainly in smaller communities this was more common than actual coin.
The first banks were established (according to the 'lies to children' version of history I was taught on this) by Knightly orders during the Crusades who had the resources (armaments, manpower, trade routes) to transport large amounts of cash around Europe. The idea being that you lodged some money with the knights in one city and could travel to another city unmolested by thieves and pick up your cash in another city on presentation of the correct credentials. This is one of the reasons cited why the monarchs of Europe were so keen to accuse the Templars of something heretical and get them all disbanded - so they could get their hands on all that gold.
A bit later in history for your needs, I think, but there was a mention in a show about the Tower of London of some gold coins the size of dinner plates which were used in the Renaissance to buy entire cargoes of goods - one of those was worth an entire ship's hold worth of cloth, for example.