Diamonds are a "Premium" resource. They are bought through real money and can be spent on pretty much anything. You can also receive Diamonds through some quests, but this is very rare, and usually not enough to be useful. A wishing well gives diamonds every once in a great while. You never find anything more than 100 in one go in-game, and even this is from a special province with special loot, for which you need to have conquered a lot of other provinces, and it's not until the Late Middle Ages.
In the Summer Event 2013 there were various chances of winning diamonds at the wheel of fortune. In the Winter Event 2013 there was only one single diamond package that could be won on one single day.
Diamonds can be spent on a lot, here are some of the things:
- Double production (motivate) in a production building or a residential building. This cannot be done if the building is motivated, and costs 25 diamonds. This can be useful when producing a day of production, but usually that is a bad idea.
- Scout a province immediately (you can't pay diamonds to have the scout scout a province though). This costs 50 diamonds, and is often worth it!
- Buying goods at a price of 5 diamonds each (so 50 lumber would cost the same as 50 whale oil - 250 diamonds)
- Buying forge points - Unless you REALLY need forge points desperately, like to go up an entire rank in a great building construction before someone else does, this is not worth it for 50 diamonds per forge point.
- Paying for missing coins or supplies, but this only applies for research! You then don't have to pay coins/supplies to unlock the technology, and you can keep them instead. This costs 60 diamonds each, and is only really good for things like Industrialization, which costs 300,000 supplies.
- Unlocking aditional slots in Military Buildings (depending on the age, this varies between 30 and 120 diamonds)
- Training military units instantly, for 50 diamonds. Generally not worth it because only Rogues, Champions and Military Drummers take longer than 4 hours to train.
- Healing and reviving units. This costs 4 diamonds per hit point. Dead units can only be revived immediately after the battle they died in. If you were using unattached units, it is a good idea to do this if they die.
- Purchasing blueprints, for 200 diamonds each. This is much better than the 2:1 trade-in because you know what you're going to get, and you can't get duplicates. Especially useful if you're just looking for one last blueprint.
- Purchasing an expansion for your city. The price for every expansion is listed in the expansion page . This is very useful since there are a finite number of expansions available through the exploration map and technology tree.
- Purchasing special buildings in each age. These buildings are far superior to all similar buildings of that age e.g. the Theater in the Bronze Age gives 540 Happiness and can be bought as soon as you reach the Bronze Age whereas the Tavern which can be bought with resources (and provides most Happiness in Bronze Age after the Theater) only provides 280 and can only be bought at the very end of the Bronze Age.
Feel free to add anything else on if I missed it off.