The six schools of Hindu philosophy or darshan are: Samkhya, Yoga, Nyaya, Vaisheshika, Vedanta, and Mimamsa. These six darshans are astika since they believe in the vedas. Over time they merged into Samkhya/Yoga, Nyaya/Vaisheshika and Vedanta/Mimamsa philosophies.
Samkhya means enumeration or number in Sanskrit and is a rational school of thought founded by Sage Kapila. It is the oldest philosophy and is based on dualism, explaining how the mind and body co-relate. It is also the source of mathematics and ayurveda, making it the basis for all Hindu philosophy. Yoga was given shape by Patanjali and is linked with Samkhya. It gives substance to the samkhya philosophy through practical experience.
The practice starts at the gross level, moves into a subtle level of the mind and then returns to the gross with a higher level of consciousness.
The difference between the two is Samkhya is a jnana yoga making knowledge the route to salvation and yoga is a dhyan yoga stressing on meditation as a route to salvation. Gauthama founded the Nyaya School, which uses the science of logic and reasoning to bring about the end of suffering.
Kannada was the founder of Vaisheshika, which is aligned with the Nyaya School and formulated the atomic structure as an explanation of existence.
Ved Vyas is the one who wrote down the oral teachings of the Vedas and Upanishads and is considered to be the founder of the Vedanta, even though the practice had been long established by other sages. Vedanta talks about the need for self-realization and oneness with the self or Brahman, as the absolute goal of life.
Jaimini founded Mimamsa as a way to provide rules for the interpretation of the Vedas and is important for the understanding of Vedanta. It is a practical explanation of dharma through rituals and obligations, which are means for enlightenment.