They fill over time, how fast depends on the champions speed.

Your question is a little bit unspecific.

I have no idea what a turn meter is.  Some skills are available for a few "turns" or "rounds" when they are used.  Does altering the turn meter effect that?  Does a turn meter do something else entirely.  Where do the creators explain these things?  I am getting frustrated.  A search of the "Guides and Tutorials" for "turn meter" returned no find.  

The turn meter is the yellow bar below your red health bar, showing how close a character is to taking their next turn. If you boost someone's TM, it means they'll get their next turn faster - if you reduce enemies' TM, it'll take longer until their next turn. It's related to speed in the sense that speed determines how quickly your TM gets filled. So if you have champions with very high speed, who moreover boost their team's TM, or reduce the other team's TM (like High Khatun, Arbiter, Lyssandra, Apothecary, etc.), it's possible that some of your toons may be taking several turns before some enemies take even one turn. 

And indeed, many skills put buffs or debuffs that last for X turns - so if for example you put a debuff on an enemy and then you keep reducing his turn meter so it takes a very long time until his next turn, it means your debuff will stay on him for a very long time, too. 

'Rounds' is something else, that refers to the three different stages of a battle - at least for content like dungeons, faction wars and doom tower, it's three times that you face new enemies with the third one usually being the boss. In arena or clan boss, there is only a single round, no matter how long the fight lasts.

Make sure you don't confuse turns with rounds! It's a very common mistake for newer players.

As for how the turn meter actually works - it's a bit of an arcane art trying to get it down properly (for whatever reason, Plarium decided not to make it simple), but the the tl;dr is that the higher your speed, the faster your turn meter will fill.

Thank you.  I had noticed some variability (mystery) regarding skills and buffs during a battle.