Kagi says: A monad is a concept that appears across multiple domains—philosophy, functional programming, spirituality, and cosmology—with different interpretations in each context.

Not a burrito?

The time-honored way of learning about monads is to write a monad tutorial. Monads provide an order dependency (that along with some hidden strictness for I/O and the like) which allows the side-effecting to be isolated from the "pure" functional side of things. The purely functional Clean language had an explicit "world" variable that you could pass around to whatever side-effecting functions you wanted. And it had uniqueness typing so that you couldn't accidentally refer to the same "world" twice. You can think about monads as being a way to hide the crufty "world". A lot of the mystery around monads comes simultaneously with a whole bunch of other newness in learning Haskell (lazy evaluation, HM type system, differing syntax, algebraic data types, currying and partial application, functional programming, etc.).

https://en.wikipedia.org/wiki/Clean_(programming_language)

I'm not a Haskell expert, but I dabbled quite a bit. This is my explanation of what monads are that doesnt rely on math lingo and understanding the maths behind it. I don't fully understand myself but maybe thats actually helpful for trying to explain to someone not familiar with the concept. My understanding relies on concepts used in other programming languages. So depending on how familiar you are with them, this might or might not be helpful for you.

A monad is kind of like a generic class to boxes that adds additional logic to the data it boxes without actually caring about the data itself.

My goto monad for that concept is a linked list and the map operator. So an instance of the linked list might be Node(5) -> Node(7) -> EmptyList. Now let's call map with a function f(x) = str(x) + " * 2 = " + str(2x). This gives us Node("5 2 = 10") -> Node("7 * 2 = 14") -> EmptyList.

Now let's separate the monad from this. The monad is the structure and logic around the data and the function that we provide. The monad doesn't care what data it holds and is doesn't care what function we provide. It only defines the structure and how functions are applied to the data it holds.