Short answers, pulled from the story.
Indian philosophy comprises six major orthodox (āstika) schools: Nyaya, Vaisheshika, Samkhya, Yoga, Mīmāṃsā, and Vedanta. There are also five major non-Vedic or heterodox (nāstika) schools: Jain, Buddhist, Ajivika, Ajñana, and Charvaka. The philosopher Vidyāraṇya identified sixteen schools in total by including Śaiva and Raseśvara traditions in his Sarva-darśana-saṃgraha.
Darśana is the traditional Vedic term for Indian philosophical traditions and means "to see" or "looking at." Classical Indian philosophers such as Chanakya preferred the term ānvīkṣikī, meaning "critical inquiry" or "investigation," used in the Arthaśāstra.
Āstika schools accept the Vedas as a valid source of knowledge, believe in the premises of Brahman and Atman, and generally affirm afterlife and Devas. Nāstika schools reject the authority of the Vedas; they include Buddhism, Jainism, Ajivika, Ajñana, and Charvaka. The orthodox-heterodox distinction is a scholarly construct found in later Indian sources, and not all sources agree on which systems belong in which category.
Charvaka, also known as Lokāyata, is an ancient school of Indian materialism that accepts only direct perception and conditional inference as valid sources of knowledge. It embraces philosophical skepticism, rejects ritualism, supernaturalism, and rebirth, and is associated with hedonistic precepts. Brihaspati is traditionally named as its founder, though the primary literature, the Barhaspatya sutras, has been lost.
Arthur Schopenhauer drew on Indian thought to develop his philosophy beyond Kant, writing in the preface to The World As Will And Representation that a reader who has assimilated the sacred primitive Indian wisdom is best prepared to understand his work. T. S. Eliot wrote that the great philosophers of India make most of the great European philosophers look like schoolboys. The 19th-century American Transcendentalist movement was also influenced by Indian thought.
Mahavira, the 24th Tirthankara, revitalised and unified Jain teachings in the 6th century BCE, roughly contemporaneous with the Buddha. Jain philosophy is distinguished by its complete separation of body from soul and its strong emphasis on ahimsa (non-violence) and anekantavada (the relativity of viewpoints). Jainism placed the strongest emphasis on non-violence of any Indian school, and these ideas influenced traditions far beyond Jainism.