Who is Sankaracharya?

Jagadguru Sri Adi Sankaracharya was the greatest exponent of the doctrine of Advaita Vedanta and a savior of Vedic Dharma. Salutations to Sankara, who is an ever shining star on the sky of Indian philosophy. 

The existence of Vedic Dharma in India today is due to Sankara. The forces opposed to Vedic religion were more numerous and powerful at the time of Sankara than they are today. Still, single-handed, within a very short time, Sankara overpowered them all and restored the Vedic Dharma and Advaita Vedanta to its pristine purity in the land pure knowledge and spirituality.

Sankaracharya occupies a very important position in the history of Indian philosophy. It can be affirmed, without any fear of contradiction, that Bharata Varsha would have ceased to be Bharata Varsha several centuries ago and would never have survived the murderous sword, the devastating fire and the religious intolerance of the successive invaders, if Sankara had not lived the life he lived and taught the lessons he taught. And those lessons are still pulsating in every cell and in every protoplasm of the true aspirant and the true Hindu. 

(excerpts taken from  http://dlshq.org/saints/sankara.htm)

What is Advaita Vedanta? 

The doctrine of advaita vedanta as expounded by Sankara can be summed up in half a verse: “Brahma Satyam Jagan Mithya Jivo Brahmaiva Na Aparah" — Brahman (the Absolute) is alone real; this world is unreal; and the Jiva or the individual soul is non-different from Brahman. This is the quintessence of his philosophy. 

According to Sri Sankara, whatever is, is Brahman. Brahman Itself is absolutely homogeneous. All difference and plurality are illusory. 

Tenets of Advaita Vedanta

• Brahman (the Absolute) is alone real; this world is unreal; and the Jiva or the individual soul is non-different from Brahman.
• The Atman is self-evident (Svatah-siddha). It is not established by extraneous proofs. It is not possible to deny the Atman, because It is the very essence of the one who denies It.
• Brahman is not an object, as It is Adrisya, beyond the reach of senses, mind or intellect. It is not another. It is all-full, infinite, changeless, self-existent, self-delight, self-knowledge and self-bliss. It is Svarupa, essence. It is the essence of the knower. It is the Seer (Drashta), Transcendent (Turiya) and Silent Witness (Sakshi). It is always the Witnessing Subject. It can never become an object as It is beyond the reach of the senses. Brahman is non-dual, one without a second. It has no other beside It.
• Sat-Chit-Ananda constitute the very essence or Svarupa of Brahman, and not just Its attributes. 
• The world is not an illusion according to Sankara. The world is relatively real (Vyavaharika Satta), while Brahman is absolutely real (Paramarthika Satta). The unchanging Brahman appears as the changing world because of a superimposition of non-Self (objects) on Self (subject - Brahman). This is called Avidya.
• The Jiva or the individual soul is only relatively real. Its individuality lasts only so long as it is subject to unreal Upadhis or limiting conditions due to Avidya. The Jiva identifies itself with the body, mind and the senses, when it is deluded by Avidya or ignorance. Just as the bubble becomes one with the ocean when it bursts, so also the Jiva or the empirical self becomes one with Brahman when it gets knowledge of Brahman. When knowledge dawns in it through annihilation of Avidya, it is freed from its individuality and finitude and realizes its essential Satchidananda nature. It merges itself in the ocean of bliss. The river of life joins the ocean of existence. This is the Truth. 
• Because samsara (or duality) exists due to ignorance or Avidya, Knowledge (Jnana) alone can make an individual realize his true nature. Karma Yoga, Bhakti Yoga, Raja Yoga etc., are necessary only to purify the individual and to help remove this Avidya. All other paths culminate in Jnana. 
• Brahma Jnana is not about acquiring any external knowledge (as Brahman can't be an object of knowledge), it just about removing the Avidya or Maya. 

(excerpts taken from the book "All about Hinduism", written by Sri Swami Sivananda)