Short answers, pulled from the story.
Advaita Vedanta is a Sanskrit compound meaning 'non-secondness' combined with 'end of the Vedas.' The prefix 'a-' means 'non-,' and 'dvaita' refers to the state in which a second thing is present, so Advaita denotes that only Brahman, the one, is ultimately real, while the phenomenal world, 'the second,' is not fully real. Vedanta refers to the Upanishads, which form the final part of the Vedic corpus.
Adi Shankara, an 8th-century CE philosopher, is generally regarded as the most prominent exponent of the Advaita Vedanta tradition. Scholars including Hajime Nakamura have concluded he was a synthesizer rather than an originator, building on earlier thinkers such as Gaudapada (7th century) and Bhartrihari (second half of the 5th century), and drawing on Buddhist philosophical methods that Gaudapada had adapted.
The four Mahavakyas, or great sentences, are: 'tat tvam asi' (That Thou Art) from Chandogya VI.8.7; 'aham brahmasmi' (I am Brahman) from Brihadaranyaka I.4.10; 'prajnanam brahma' (Prajnanam is Brahman) from Aitareya V.3; and 'ayamatma brahma' (This Atman is Brahman) from Mandukya II. The Advaita tradition treats these statements literally as direct revelations of the identity of the individual self and Brahman.
In Advaita Vedanta, maya refers to the appearance or illusory manifestation of the phenomenal world. The notion that the world is illusory was introduced by the 13th-century scholar Prakasatman, founder of the Vivarana school, and is often misattributed to Adi Shankara himself. According to Paul Hacker, maya was not a prominent theme for Shankara and 'the word maya has for Shankara hardly any terminological weight.'
The Bhamati school, founded by Vachaspati Misra (a student of Mandana Misra), holds that the mahavakyas yield indirect knowledge that must be made direct through deep and continuous contemplation, and locates avidya (ignorance) in the individual self. The Vivarana school, founded by Prakasatman around the 13th century, follows Shankara's subitist position that hearing the mahavakyas directly causes liberating knowledge, and locates avidya in Brahman itself.
In the 19th century, the influence of Vidyaranya's Sarvadarshanasamgraha and Orientalist Indologists including Paul Deussen caused Western scholarship to overemphasise the importance of Advaita Vedanta, treating it as the paradigmatic example of Hindu spirituality despite the numerical dominance of theistic, bhakti-oriented religiosity among Hindu practitioners. Paul Hacker later described the Indian response to this colonial framing as neo-Vedanta.