So I just received word that one of my horses was sent to the Humane Something or Other and I take it to mean it was because I missed payment at the stable I was boarding with, but what does that mean exactly? Is there a way I can get the horse back?

You can go to the horse's pages and reclaim them :). There will be a link for you to do that.

A boarding stable/kennel has the option to send animals to the humane alliance after someone hasn't paid for a certain amount of time.