Adi Shankara's treatises on the Upanishads, the Bhagavad Gita and the Brahma Sutras are his principal and almost undeniably his own works. Although he mostly adhered to traditional means of commenting on the Brahma Sutra, there are a number of original ideas and arguments. He taught that it was only through knowledge and wisdom of nonduality that one could be enlightened.
Shankara's opponents accused him of teaching Buddhism in the garb of Hinduism, because his non-dualistic ideals were a bit radical to contemporary Hindu philosophy. However, it may be noted that while the Later Buddhists arrived at a changeless, deathless, absolute truth after their insightful understanding of the unreality of samsara, historically Vedantins never liked this idea. Although Advaita also proposes the theory of Maya, explaining the universe as a "trick of a magician", Shankara and his followers see this as a consequence of their basic premise that Brahman is real. Their idea of Maya emerges from their belief in the reality of Brahman, rather than the other way around.
Shankara was a peripatetic orthodox Hindu monk who traveled the length and breadth of India. The more enthusiastic followers of the Advaita tradition claim that he was chiefly responsible for "driving the Buddhists away". Historically the decline of Buddhism in India is known to have taken place long after Shankara or even Kumarila Bhatta (who according to a legend had "driven the Buddhists away" by defeating them in debates), sometime before the Muslim invasion into Afghanistan (earlier Gandhara).
Although today's most enthusiastic followers of Advaita believe Shankara argued against Buddhists in person, a historical source, the Madhaviya Shankara Vijayam, indicates that Shankara sought debates with Mimamsa, Samkhya, Nyaya, Vaisheshika and Yoga scholars as keenly as with any Buddhists. In fact his arguments against the Buddhists are quite mild in the Upanishad Bhashyas, while they border on the acrimonious in the Brahma Sutra Bhashya.
The Vishistadvaita and Dvaita schools believed in an ultimately saguna Brahman. They differ passionately with Advaita, and believe that his nirguna Brahman is not different from the Buddhist Sunyata (wholeness or zeroness) — much to the dismay of the Advaita school. A careful study of the Buddhist Sunyata will show that it is in some ways metaphysically similar as Brahman. Whether Shankara agrees with the Buddhists is not very clear from his commentaries on the Upanishads. His arguments against Buddhism in the Brahma Sutra Bhashyas are more a representation of Vedantic traditional debate with Buddhists than a true representation of his own individual belief. (See link: Shankara's arguments against Buddhism)
Dvaita, a school of Vedanta (the most widespread Hindu philosophy) founded by Shri Madhvacharya, stresses a strict distinction between God and souls. According to Madhva, souls are not created by God but depend on Him to evolve or transform. Satguru Sivaya Subramuniyaswami explains dualism in his book, Dancing with Siva, by stating that in dualism, God is seen as the efficient cause of the universe and not as the material cause. He is the potter causing the clay to emerge, rather than serving as the source of the clay itself.
Like Ramanuja Shri Madhvacharya espoused a Vaishnava theology that understands Brahman to be endowed with attributes and a personal God, Vishnu. By Brahman, he referred to Vishnu, as per his statement (brahmashabdashcha vishhnaveva) that Brahman can only refer to Vishnu. Madhva states that Vishnu is not just any other deity, but is rather the singular, all-important and supreme one. Vishnu is always the primary object of worship, and all others are regarded as subordinate to Him. The deities and other sentient beings are graded among themselves, with Vayu, the god of life, being the highest, and Vishnu is eternally above them.
Dvaita, or Dualistic philosophy, also known as Bheda-vâda, Tattva-vâda, and Bimba-pratibimba-vâda, is the doctrine propounded by Ananda Tîrtha (also known as Madhvacharya) that asserts that the difference between the individual soul or jîva, and God,(Îshvara or Vishnu), is eternal and real. Actually, this is just one of the five differences that are so stated -- all five differences that constitute the universe are eternal.
The Dvaita doctrine was summarized by Vyasa Tirtha as comprising nine tenets or prameyas.
Vishisthadvaita is a qualified monism in which God alone exists but admits plurality. By holding such beliefs, Vishisthadvaita is midway between Advaita and Dvaita. The book, "All about Hinduism" by Swami Sivananda discusses aspects of Vishisthadvaita, a theological school whose most important proponent was Ramanuja.
In more specific terms Vishishtadvaita conclusions may be briefly summarised as below.
The Absolute Supreme Reality referred to as Brahman, is a Transcendent Personality with infinite superlative qualities. He is Lord Vishnu, also known as Narayana. He creates the other two members of the Trimurti, namely, Creator Brahma and Shiva, the Lord of Deluge.
Narayana is the Absolute God. The Soul and the Universe are only parts of this Absolute and hence, Vishishtadvaita is panentheistic. The relationship of God to the Soul and the Universe is like the relationship of the Soul of Man to the body of Man. Individual souls are only parts of Brahman. God, Soul and Universe together form an inseparable unity which is one and has no second. This is the non-duality part.
Matter and Souls inhere in that Ultimate Reality as attributes to a substance. This is the qualification part of the non-duality.
Souls and Matter are only the body of God. Creation is a real act of God. It is the expansion of intelligence. Matter is fundamentally real and undergoes real revelation. The Soul is a higher mode than Matter, because it is conscious. It is also eternally real and eternally distinct. Final release, that comes, by the Lord's Grace, after the death of the body is a Communion with God. This philosophy believes in liberation through one's Karmas (actions) in accordiance with the Vedas, the Varna (caste or class) system and the four Ashramas (stages of life), along with intense devotion to Vishnu. Individual Souls retain their separate identities even after moksha. They live in Fellowship with God either serving Him or meditating on Him. The philosophy of this school is SriVaishnavism, a branch of Vaishnavism.
The succession of great Master-Expositors and spiritual giants of Vishishtadvaita school starts with the twelve Alwars, who left behind an imperishable legacy of Tamil devotional poetry in the form of 4000 songs, now called the Nalayira-prabandham.