Short answers, pulled from the story.
The two major traditions are Hindustani music from North India and Carnatic music from South India. Hindustani music emphasizes improvisation and exploration of all aspects of a raga, while Carnatic performances tend to be shorter and composition-based. The two systems were not considered distinct until around the 16th century, after the socio-political upheaval of the Delhi Sultanate era separated north from south.
Raga and tala are the two foundational elements of Indian classical music. A raga is a melodic framework built from a set of notes in a specific order designed to evoke a particular mood or rasa in the listener. Tala is the metrical time cycle that measures rhythm; it does not follow the simple strong-weak beat patterns of Western music but allows accents to be shaped by the musical phrase itself.
Purandara Dasa (1484-1564) is called Pithamaha of Carnatic music, a Sanskrit term meaning great father or grandfather. A Hindu composer and monk who lived in Hampi in the Vijayanagara Empire, he systematized classical Indian music theory and developed a teaching methodology called Suladi Sapta Tala that remains in use today.
The Sangita Ratnakara, written by Sarngadeva around 1230 CE under the patronage of King Sighana of the Yadava dynasty in Maharashtra, is regarded as the definitive text by both the Hindustani and Carnatic traditions. It identifies seven tala families, subdivides them into rhythmic ratios, and presents a systematic methodology for improvisation and composition.
Tansen studied music and introduced innovations for roughly the first sixty years of his life under the Hindu king Ram Chand of Gwalior, then performed at the Muslim court of Akbar during the 16th century. Many musicians consider him the founder of Hindustani music, and many modern gharanas, the teaching lineages of Hindustani music, trace their roots to his style.
Indian classical music gained a following in the United States after Ravi Shankar arrived in 1958 and began recording albums. By 1967, Shankar and other artists were performing at rock music festivals, and Shankar performed at Woodstock in 1969 before an audience of over 500,000. This wave of interest lasted into the mid-1970s.