Medals can be won in tournaments after unlocking the corresponding towers in the world map. Their main purpose is to unlock city extensions.
To win medals, you need to unlock the province of the continent (in the world map) that contains a tournament tower, then you have to fight and defeat your neighbors to gain points.
Every Sunday, the tournament closes and winners obtain their medals, the amount of which depends on the age the tower corresponds to.
You may choose to attack neighbors, support them with aid, or do nothing. For supporting them, you are given a token amount of 20 coins.
Medals could also be traded with Leonardo Da Vinci for blueprints during the Feb 2013 event.
Medals can also be earned by helping others develop the Great Buildings.
Medal Scoring in PvP Tournament Tower
The number of medals needed to unlock a medal expansion increases with each expansion that has already been purchased. The first one costs 10 medals, the second costs 50 medals, and so on. The cost of each medal expansion can be viewed here.