Wikipedia mentions the following on the six orthodox systems:
Several Indian intellectual traditions were codified during the medieval period into a standard list of six orthodox systems or ṣaḍdarśanas, all of which cite Vedic authority as their source.Nyaya, Vaisheshika, Samkhya, Yoga, Mimāṃsā and Vedanta are classified as āstika schools:
Nyāyá, the school of logic
Vaiśeṣika, the atomist school
Sāṃkhya, the enumeration school
Yoga, the school of Patañjali (which assumes the metaphysics of Sāṃkhya)
Mimāṃsā, the tradition of Vedic exegesis
Vedanta or Uttara Mimāṃsā, the Upaniṣadic tradition.
These are often coupled into three groups for both historical and conceptual reasons: Nyāyá-Vaiśeṣika, Sāṃkhya-Yoga, and Mimāṃsā-Vedanta.
The following is an excerpt from an essay about the six houses of philosophy. Read here for the complete essay.
Nyaya - The Nyaya school was founded by sage Gautama. Sixteen major topics were discussed in this system, the most important of which is pramana, the source of valid knowledge. Actually, Nyaya is a school of logic, and all other schools of Indian philosophy use the Nyaya system of logic, in whole or in part, as a foundation for philosophical reasoning and debate. Navya-Nyaya or Neologic, a further development of this school, occurred in the 16th century in Bengal and Mithila.
Vaisesika - Kannada is the founder of this school, which is associated with the Nyaya system. This school discusses seven major topics: substance, quality, action, generality, uniqueness, inherence and non-existence. This school is called Vaisesika because it considers, uniqueness, as an aspect of reality and studies it as a separate category. Under the topic of substance, it deals with the physics and chemistry of the body and the universe. The theory of atomic structure was established by this school. Its practical teaching emphasizes dharma, the code of conduct that leads man to worldly welfare and to the highest goal of life.
Samkhya - Kapila is traditionally cites as the founder of this school, although his Samkhya Sutras have been lost. The Samkhya-karika of Isvarakrsna, the oldest text on this philosophy, cites the name of Kapila, Asuri and Pancasikha as previous teachers of this school. It is considered to the oldest of the philosophical systems.
Samkhya is a dualistic philosophy that believes in the coexistent and interdependent realities, conscious Purusha and unconscious Prakrti. Purusha is ever pure, wise and free but it becomes a subject of pain and pleasure when it identifies itself with Prakrti. Prakrti is the material cause of the universe and is composed of three gunas – sattva, rajas and tamas that correspond to light, activity and inertia respectively. The state in which the gunas are in equilibrium is called Prakrti but when disturbed the state is called Vikrti. Disturbance of the equilibrium of Prakrti produces the material world, including the mind, which is supposed to be the finest form of material energy.
Samkhya philosophy explains the dynamics of the body and nature of mind. It is the mother of mathematics as well as Ayurveda and is indeed the very basis of Eastern philosophy.
Yoga - Yoga and Samskhya are allied systems. Although Yoga philosophy was known even in the Vedic and pre-Vedic periods, it was not formally systematized until it was codified by Patanjali in about 200 BC. The Yoga Sutras contain 196 aphorisms, which are divided into four sections. Yoga studies all aspects of human personality and teaches one how to control the modifications of the mind through practice of meditation and detachment and surrender to higher consciousness. It prescribes a holistic system of practice beginning with the yamas and niyamas (ethical and behavioral codes) and proceeding through the asanas (physical postures), pranayama (breathing exercises), pratyahara (control of senses), dharana (concentration), dhyana (meditation) and culminating in samadhi. In this system the individual self is the seeker and pure consciousness is the ultimate reality that he finds within. Practicality is the main feature of this system.
Mimamsa - Jamini was the founder of this system that accepts the Veda as the final authority on all questions. It provides a comprehensive method for interpreting and understanding the underlying meaning of the Veda. It lays great emphasis on rituals, worship and ethical conduct and provides a systematic lifestyle and direction. Mimamsa offers guidelines for practical application of Vedantic theory. This school is foremost in the analysis of sound and mantra.
Eventually this school was divided into two groups: the school founded by Prabhakara and the one founded by Kumarila Bhatta. According to the former there are five sources of valid knowledge: perception, inference, comparison, testimony and postulation. According to the latter there is only one source of knowledge – noncognition.
Vedanta - was taught and practiced by the sages of the Vedas and Upanishads and was handed over through a long line of sages. But Veda Vyasa, who codified these teachings in the Brahma Sutras, is considered its founder. Until the time of Sankara, Vedanta was mainly transmitted through oral tradition but sometime between the 6th and 8th centuries a.d. Sankara reorganized the system of this monistic school of thought. After him numerous teachers wrote commentaries on the Brahma Sutras, interpreting it in various ways and thus establishing various schools within the single system of Vedanta.
The major schools of Vedanta are Advaita (nondualistic), Dvaita (dualistic), Dvaitadvaita (both dualistic and non dualistic), Visistadvaita (qualifies nondualism) and Visuddhadvaita (pure non-dualism). Of these schools Sankara’s Advaita and Ramanuja’s Visistadvaita are the most important. Sankara’s Advaita Vedanta covers all the other systems. The main teachings of Vedanta is that self-realization is the actual goal of life, that the essence of the self is the ever existent consciousness and bliss, the Self is free from all qualifications and limitations, that the self is essentially Brahman, supreme consciousness and this Brahman is the absolute, transcendent, attributeless reality but it eternally embodies itself within itself the capacity or power called maya, which is the basis of mind and matter.