What does "shards per turn" mean?

Its meaningless wording. In general all their stats have a per turn.

You can think of each altar as producing 1 of the 'shard' resource per turn, the main difference* between these shards and other resources is that your shard count is not cumulative, i.e. it doesn't add up from one turn to the next, but resets every turn. So each turn your 3 earth altars produce 3 of the earth shard resource, and next turn you lose those 3 and your altars generate another 3 (or 2 or 4 if you've lost or gained an altar since last turn).

It's actually not the only resource that functions like this - say food or production are also noncumulative, you get a certain amount each turn which you can either use or waste as the case may be, there's no 'saving' up the extra. From a programmer's perspective, it makes sense to have every resource generated on a per turn basis, and simply flag whether you want them to add up from one turn to the next or reset each time. It allows you to write one piece of code that can deal with any resource, it just behaves slightly differently depending on that flag.

But yeah, from a user perspective, that all probably seems bizarre - it'd be easier to understand if they added a little interface translation to represent 'shards per turn' as simply 'shards'.

*Well actually, there's another difference, shards are not consumed when 'used' by a spell. This is not necessarily connected to the whole accumulation thing discussed above, though. Influence, for example, accumulates from one turn to the next, but it's not always consumed when 'used', it has an effect on the way AIs treat you without being used up - the same way shards have an effect on spells without being used up. So each resource can either accumulate from one turn to the next or not, and it can be consumed when used or not (or both, if it has multiple uses like influence does).

Okay, thanks for the explanation.  I just wanted to make sure I wasn't missing some functionality of the shards.  So they do not accumulate turn-to-turn, nor are they used up in spell casting.  They are simply a total number of shards held by a faction.

I guess in one sense, to a player, per-turn makes sense when you upgrade the altars (which I have not done yet in my game even), which I assume will eventually yield more than one shard each.  Then maybe there is a sense of being able to "extract" more shards, and so a frequency of sorts (per turn) makes sense.

This assumption is incorrect. All that upgrading shard altars does is increase the mana production of the shard altar.

As far as I know, there is no existing way in the unmodded game to increase the number of shards you have without building/capturing another shard altar, taking certain level-up choices for high level conclaves, or building certain unique structures in the Magic tree. There might also be a quest or two that allows this, but I'm not certain. There is a spell that converts shards of any type to shards of one type, but I don't greatly care for that spell since the shards it makes (Death Shards) are not shards that I consider to be particularly useful.

I believe each (or most) of the theme areas (Wildlands, Scap, etc) have unique resource which generates a shard once you've built a monument on it.

Good to know.  Thank you.

Elemental temples (an upgrade from the elemental shrines you were mentioning) give both an extra shard and 3 extra mana per turn, you need to research Immortal Codex to unlock them.