'There are a significant number of concepts that aren't easily explained in plain english required to get at the heart of "what is a monad?".'
Indeed, it's arguably not even the right question. "When is something monadic?" is a better one; "monad" is an adjective, not a noun. In Java terms, "Monad" is an interface and something is "monadic" if it implements the interface. (Except Java's type system can not express "Monad", so that doesn't work as a concrete example, but the idea is there.)
I'm sure you've seen me claim this on /r/haskell, where I think people operate in a context where they find this so obvious that the very idea that there could be a problem is confusing... this is the sort of context that leads me to that claim, though. Regardless of whether it's a noun in math, in programming it's an adjective.
Indeed, it's arguably not even the right question. "When is something monadic?" is a better one; "monad" is an adjective, not a noun. In Java terms, "Monad" is an interface and something is "monadic" if it implements the interface. (Except Java's type system can not express "Monad", so that doesn't work as a concrete example, but the idea is there.)
I'm sure you've seen me claim this on /r/haskell, where I think people operate in a context where they find this so obvious that the very idea that there could be a problem is confusing... this is the sort of context that leads me to that claim, though. Regardless of whether it's a noun in math, in programming it's an adjective.