Advaita, Vishishtadvaita and Dvaita are three schools of Vedanta reading the same texts, the Upanishads, the Bhagavad Gita and the Brahma Sutras, and arriving at three incompatible answers to a single question: what is the relation between the self and Brahman? The differences are precise, and a vague account of them is worse than none.
Shankara: identity
For Advaita, the self is Brahman. Not a part of Brahman, not like Brahman, not united with Brahman, identical, and always was. The plurality of selves and the world of objects is the work of avidya, ignorance, which superimposes distinction where there is none, in the way a rope is taken for a snake. Shankara is careful about what this does and does not claim: the world is not a private hallucination, it is real at the transactional level and unreal only from the standpoint of the absolute. Liberation therefore accomplishes nothing and goes nowhere. It removes an error. The final Brahman is nirguna, without attributes; the personal God who is worshipped is Brahman as it appears from inside ignorance, which is why Advaita can sustain a full devotional life and still call devotion provisional.
Ramanuja: identity-in-difference
For Vishishtadvaita, that account destroys everything it touches. Ramanuja argued that an undifferentiated reality cannot be known, spoken of, or taught, so a scripture teaching it would be self-defeating; and that if the world were unreal, so would be the bondage from which liberation liberates. His alternative: reality is one, and internally differentiated. Selves and matter are real and eternally distinct from one another, and they stand to Brahman as a body stands to the self that indwells and directs it. Brahman is the inner controller of all of it. The world is God's real body, not an appearance. So the self is neither identical with God nor separate from him: it is a real mode of him, and in liberation it comes to share his knowledge and bliss while remaining itself. Ramanuja's God is personal, has all auspicious qualities, and is characterised above all by accessibility, which is why the route is bhakti, and prapatti, surrender, for those who cannot sustain it.
Madhva: difference
For Dvaita, Ramanuja stopped halfway. Madhva held difference to be not a problem to be managed but the structure of what exists, and he enumerated it: five eternal differences, between God and each self, God and matter, one self and another, self and matter, and one material thing and another. Only Vishnu is svatantra, independent; everything else is paratantra, dependent. The self is a reflection or image of God and is never identical with him, before or after liberation. Madhva went further than anyone else in the tradition in two respects. He held that selves are intrinsically graded, so that liberated souls enjoy different degrees of bliss according to their natures; and he taught that some souls are fit only for darkness and are eternally damned, a doctrine with almost no parallel in Hindu thought, and the position his opponents attacked most often.
What is actually at stake
The quarrel is not scholastic hair-splitting. It decides whether the goal of religious life is knowledge that dissolves a mistake, or love that reaches a person who remains other; whether devotion is a means to be transcended or the final state itself; and whether the world you are standing in is the appearance of something else or the body of God. Three schools, three answers, and about twelve hundred years of argument that has never been settled and was conducted at a level of technical rigour that repays reading.