This app was mentioned in 2 comments, with an average of 4.00 upvotes
There is the EURik app for Android (and maybe iPhone, I dunno), which lets you catalogue your Euro coin collection and has mintages in different types. You can find it here: https://play.google.com/store/apps/details?id=com.osumsky.eurik
But I'll still try to explain in this comment the general basis:
-Firstly, every country has its own national design on the regerse (back) of the coins, which is minted on the normal circulating coins, while the obverse has a common image depicting the value of the coin
-But every country can mint up to 2 commemorative coins every year, to celebrate important events, people and landmarks of the nation. Most of the time, these coins are only related to a specific country, but sometimes the whole EuroZone decided to mint specific commemorative coins to celebrate important events for the EuroZone (like, for example, the coin that celebrated the 10 year anniversary of the euro)
-But that's not it: every country can produce every year additional coins that are really really special, for various reasons:
--They are made with special metals and metal alloys, and depicts really specific events that wouldn't be possible to celebrate otherwise (some can even glow in the dark!)
--They aren't circulating, which means that you won't find them in your change or at the supermarket, unless some incredibly unaware guy decides to buy one from the mint (which is basically the only way to get them) and releases it in circulation (which apparently is not that rare in France?)
--They can have special values, like 1/4€ (the first of this kind ever created), 5€, 10€, 20€, 100€... The highest-value one was produced by Austria to celebrate the 100th anniversary of the National Philarmonic, which has a whopping value of 100k€!
--N.B.: these coins are legal tender only in the state that produced them, for example if you decided to use for some dumb reason the 100k one (which is, FYI, made of pure 99.9999999%, or something like that, gold), you could only use it in Austria
Hope this cleared up aaaaaaall of your doubts!
My favorite coin is the 2€ from Malta. I also like the ones from Italy, because overall they have some of the best looking motifs that also are unique for each denomination (and therefore, interesting).
By the way, for people that are interested, I wanted to recommend two Android apps that are listing all the coin designs: EURik - ordered by country, great overview over the different series, good for collecting; EuroCoins - ordered by coin value, great for just browsing through the coin designs, including info about the motif.