Six Schools of Indian Philosophy

Yoga	Started by Patanjali Practical methods for direct experience:  Yoga systematically deals with all of the levels of one's being, striving to experience the eternal center of consciousness.  Yoga is best described in the Yoga Sutras and involves systematic witnessing of your inner states, so as to experientially go beyond all of them to the center of consciousness.  Yoga is often called Sankhya-Yoga, as Yoga contains the practical methods to realize in direct experience the truths of Sankhya philosophy
Sankhya	Started by Kapil Muni Framework of manifestation:  Sankhya philosophy offers a framework for all the levels of manifestation, from the subtlest to the grossest.  Sankhya comes from samyag akhyate, which literally means that which explains the whole.  Sankhya deals with prakriti (matter), purusha(consciousness), buddhi or mahat (intelligence), ahamkara (I-am-ness), three gunas (elements of stability, activity, and lightness), mind (manas), cognitive and active senses (indriyas), and the five subtle and gross elements (earth, water, fire, air, and space). In light of its breadth, it contains all of the domains of Vaisheshika, Nyaya, and Mimasa, which are described below.
Vedanta	Based on Upnaishads (Shankaracharya is founder) Contemplative self-inquiry:  Vedanta philosophy and practice provides contemplative methods of self-inquiry leading to the realization of one's true nature, that which is not subject to death, decay, or decomposition.  A major key of these practices is contemplation on the Mahavakyas.  The teachings of Vedanta are best captured in the books of the Upanishads.  The text, Vivekachudamini (Crest Jewel of Discrimination) by Adi Shankaracharya is an excellent source.
Vaisheshika	Started by Prashastapada Physical sciences:  The Vaisheshika system emphasizes the physical sciences such as chemistry.  It includes exploring the elements of earth, water, fire, air and space, as well as time, mind and soul.  Focus is on salvation through recognition of atomic nature of universe.
Nyaya	Started by Gautam Rishi  Reasoning:  The Nyaya system deals with logic, the process of reasoning.  Doubt is considered a prerequisite for philosophical inquiry.  Other systems of Indian philosophy draw on this process.
Mimasa	Started by Jaimini Freedom through action:  The Mimasa system pursues freedom through action.  It has a detailed philosophy related to ritual, worship and ethical conduct, which developed into the philosophy of karma.