Monism holds that reality is ultimately one, one substance, one principle, or one kind of thing. Its instances include Advaita Vedanta, for which only Brahman is finally real; Spinoza, for whom there is a single substance with infinite attributes, called God or Nature; Stoic cosmology, with its one rational principle pervading everything; and, in an entirely different register, the materialist monism that holds everything to be one kind of stuff.
Dualism holds that there are two irreducible principles, and the word covers several quite different claims whose conflation is the standard error in this area. Metaphysical dualism of good and evil, in Zoroastrianism and more radically in Manichaeism, posits two opposed cosmic powers; it is the position that most cleanly solves the problem of evil, at the cost of divine omnipotence, which is precisely why the monotheisms rejected it so firmly. Samkhya's dualism of purusha and prakriti divides consciousness from everything else and has nothing to do with good and evil. Cartesian dualism of mind and body divides thinking substance from extended substance and generates the interaction problem that has occupied philosophy of mind ever since. These three have almost nothing in common beyond the number two.
Nondualism, advaita, is the position that most needs distinguishing from monism, and its adherents insist on the distinction. Advaita means "not two" rather than "one", and the negative formulation is deliberate: to assert that all is one is still to make a positive claim about a totality, and still to count. Advaita Vedanta holds that the apparent duality of self and Brahman is not real, without thereby asserting a monistic substance in the way Western monism does. Much of Mahayana Buddhism, and Madhyamaka especially, is nondual in a stronger sense still, refusing "one" and "two" and every other position, since emptiness is not a thing that everything turns out to be. Kashmir Shaivism, Dzogchen and the Sufi wahdat al-wujud are generally placed here as well.
The distinctions matter practically, because a different path follows from each. If reality is one substance, knowledge of it is the aim. If there are two irreducible principles, discriminating between them is the aim, which is exactly what Samkhya prescribes. If duality is an appearance, then neither uniting nor separating is required, but only the ending of a misperception, which is why Advaita makes knowledge rather than action the means.
Western scholarship's long habit of translating advaita as "monism" has been criticised at length by scholars of Indian philosophy on precisely this ground, and the criticism is a good illustration of how much a translation choice can decide in comparative work before any argument begins.