Coins were exclusively used as a means of exchange between strangers. Because most transactions would take place between people in the same town or village, coinage, up to the rise of the more 'anonymous' cities like Bruges or Paris in the very latest Medieval period, was not widely used at all except amongst travellers, soldiers, aristocrats etc. This had as a consequence that coinage was associated specifically with authorities and the state, helped along with the symbology on the coins themselves (portraits of rulers, anonymous religious images). This meant in turn that the boundary between coins and amulets, for example, was much more blurred than it is today.

So in short, yes, common people would use coins very differently from us today, but traders and merchants would use them quite similarly to us.

Medieval numismatics is an entire field on its own. For the Viking Age, see for example Dagfinn Skre's work on Kaupang: Means of Exchange. Dealing with Silver in the Viking Period, or for the Frankish world Frans Theuws' work like Peasant agency, long distance trade and the early medieval economy. More generally see Ian Hodder's Dark Age economics, a new audit (2012), or the very controversial (and often inaccurate, but inspirational) work of David Graeber, Debt, the first five thousand years.

Big, big question. The Middle Ages spanned an incredibly diverse chronology.

In order for coins to have any sort of utility, they need to have some sort of entity enforcing that utility, i.e. a government. In the early Middle Ages, there were many governments capable of minting coins and enforcing their value (this includes protection from problems like counterfeiters) and a market where coins represent a useful means of exchange.

Take, for example, the Sutton Hoo burial. The Sutton Hoo man was discovered with a number of Merovingian coins. These objects held some sort of symbolic value representing the relationship between this East Anglian king and the major monarchy in western mainland Europe.

Contrast this with 13th century Italy. At this point, we find numerous governments of city-states whose urban populations are greater than anywhere else in Europe except for the low countries, and their economies are based on production and exchange. Here, coins as modes of exchange is more than viable, it's necessary. Thus, the more powerful city-states produce coins that are used in small scale transactions. At various points both the Florentine and Venetian pennies were in wide circulation in this part of the world.

As /u/Aerandir observes, coins were in far less common use than today. Partof the reason is the absence or scarcity of small denominations for everyday local purchases: in England one silver penny would get the average person through the day even after the inflation of the 13th century, and there was nothing smaller until the advent of the halfpenny (12th century) and farthing (quarter-penny, 13th century), both seemingly minted in small quantities (Edward I's reported four million farthings sound a lot but only came to £4-5,000 in an economy producing around £4-5 million annually). The arrival of gold coinage in the 14th century was no help, each gold "penny" equalling twenty silver ones. One solution was to cut the silver coins up, another was barter or payment in goods for work. But not until the silver inflow, inflation and debasements of the 16th century did coins start to become suited to everyday exchange.

Although there are lots of thorough answers here already, I just want to add a point that doesn't seem to have been mentioned: coins are a standardized quantity of a precious metal. If you know how much gold one bushel of wheat is worth, you know how much how many gold coins it is worth. Because humans ascribe value to precious metals, this allows quantities of precious metals to be used as a store of value.

Here's an example: A Florentine gold Florin in the 13th century is more or less consistently 3.5368 grams of 24-karat gold. Silver Florins were issued and were worth one-tenth of a gold Florin. A Venetian Gold Zecchino was also of similar weight (around 3.5 grams). Half-value Zecchini and quarter-value Zecchini were also issued.