The Upaang - Darshan Shastra is the literature in which the Rishis broadly described the philosophy from the Vedas through analysis. There are six schools of thought, known as Vedic Darshans, based on the Vedas, which are the foundation for all six. These six schools are Nyaya, Vaisheshika, Saankhya, Yoga, Meemaamsaa, and Vedaanta.
Nyaya, authored by Rishi Gautama ji, focuses on attaining Moksha (Salvation) by acquiring philosophical knowledge of 16 objects, including Proof/Evidence, Proposition, and Logic. Its main theme is to examine things through proofs and evidence. The order in which the 16 objects are explained is Name, Features, and Detailed examination. An ancient and authentic commentary on this Darshan is available by Maharishi Vaatsyayan ji.
Vaisheshika, authored by Rishi Kanaad ji, describes the true form of Dharma and how to achieve worldly and spiritual success through living a righteous life. Its main themes are "Nothing can be created without any material" and "Absence of the cause is absence of the effect." It provides a true conception of six entities, including Noumenom, Attribute, and Action, which is helpful for attaining Moksha.
Saankhya, authored by Rishi Kapil ji, deals with Prakriti and its products and Purusha. The word "Purusha" denotes both God and the Soul. The basis of this Darshan is Sat Kaarya Vaad "Nothing can ever become something, nor can something ever become nothing." It explains the order of creation and dissolution of the universe from Prakriti. Moksha is obtained only through discrimination gained by knowing the true form of the material world and Purusha.
Yoga, authored by Maharishi Patanjali ji, deals with Saadhana, Dhyan, Samaadhi, and provides a clear-cut idea about God, Soul, Material world, true form of God, Vedic worship, and means of obtaining Moksha. It also explains subtle topics like what yoga is, the reason for the soul's bondage, the different stages and attainments of the yoga practitioner, and the fluctuations of the mind and how to stop them.
Meemaamsaa, authored by Rishi Jaimini ji, discusses the science of morals in detail, with the concept of Dharma, Dharmi, and Yagya. It describes the duties and non-duties of mankind, ranging from family life to national service, through which the entire nation can develop. The Yagya (Sacrifice) is the great form of Dharma.
Vedaanta, authored by Rishi Vyaasa ji, deals with Brahmaa (Iswhar) and the attainment of Brahmaa/Moksha. Its other names are Brahmasootra and Uttar Meemaamsaa. Brahmaa is omnipotent, omnipresent, and blissful, free from the sorrows of birth and death. Jeevaatmaa (Soul) is a different entity having little intellect and subtle. God creates the universe from Prakriti. The meaning of Vedaanta is "The essence of the Vedas."
In summary, some common principles of the six Darshans are that Moksha is obtained by freedom from the three types of sorrows, God, Soul, and Prakriti are the eternal causes of the creation of the universe, the nonexistence of existent objects and existence of nonexistent objects is impossible, and Vedas are self-proven as the gospel-word of God. The Soul is inconsequential and different from the body and mind