Monism is when you're looking at a cookie, and you're not looking at crumbs. You're not looking at butter, flour, salt, and sugar, or at the act of mixing, a mould (cutter), or the heat of an oven that was used to make it. Monism is us when we're just looking at a cookie, and that's all we see, is a cookie. And we say, "Oh, look. It's a cookie!" It has parts--if we switch mental modes then we know it has parts, and it has process by which it was made. But in seeing only a cookie, in the monist perspective, we are genuinely not looking at the rest. We see The One and we consider it to be the truth.
That's how a human is made, by the way. This guy in our heads that we treat like a "being," that we believe is real and aware to the extent that we give it rights, that we elevate so high as to build whole cities, countries and civilizations around, and that we give to "live on" after our body has expired. It's a token man, made of neurons, memories, and circumstances. In our monism, we consider it the truth.
If you get a chance to read Zen and the Art of Motorcycle Maintenance, a motorcycle works because it's put together well. But it's not just its physical parts, it's also the manufacturing, the maintenance, and a whole lot of skill that goes into keeping it working. Those are parts too. All its parts and processes are needed for the whole to exist. But the motorcycle is also the billion year old plants that died to put fuel in the engine and the metals that were formed along with the creation of the Sun and the Earth. In this sense the whole world is parts of the motorcycle. The circumstance that has resulted in this moment in time, in this place, on this planet, in this universe created a motorcycle. It created you too.
Where monism is a perspective on the world, non-dualism does away with perspective. Opposites are an arbitrary divide. An opposite would be that there is a cookie and there are non-cookies (cakes, pastries). There is black, and there is non-black. There is truth, and there is non-truth. There is, and there is not. Dualism. I have in mind Alan Watts' example of a window outside of which is a tree. Through the faculty of a window we separate ourselves from what we see "outside" the window, and we subsequently consider ourselves to be an "inside." i.e. a mind. Non-dualism dumps the unnecessary part of that picture, which is the window.
IMO, it was added by us, by language to be precise. It's an illusion of thought/speech. We are all passively taught the think in terms of the window from the moment we learn our first words, and it is passed down through generations in language and culture. We are taught that there is a subject "I" that acts and interacts (verbs) with objects (nouns) that are non-"I." Doing away with that window doesn't make you a tree, but it does put things in a whole new perspective.