While the Vedas are considered the most sacred and treasured texts of India, it is the Upanishads that transferred the wisdom of the Vedas into practical and personal teachings. The word Upanishad literally means "sitting down beside” and the collection of Sanskrit texts known as the Upanishads are thought to be the direct teachings received at the foot of the ancient Indian sages. In these sacred texts we see an internalization of the sacrifice and worship extolled in the Vedas and a deeper understanding and exploration of the internal world of mind and spirit.
Composed over several centuries and in many volumes, the Upanishads reflect a strong need to express and communicate the deep mystical states and spiritual contemplations that the ancient yogis experienced. According to tradition, there were over two hundred Upanishads, but there are only eleven "principal" Upanishads, as commented on by the ancient sage Shankara. The texts are written in a passionate poetic verse describing mystical states and spiritual concepts or in descriptive short stories and dialogues between historical figures.
The teachings of the Upanishads revolve around four primary spiritual themes. The first and most important is the realization that the ultimate, formless, and inconceivable Brahman (Godhead) is the same as Atman, as our internal soul. Brahman represents the entire universe, and the Atman is a little piece of that divine oneness that we contain inside us. This philosophical idea is summed up in the mantra Tat Tvam Asi (That Art Thou). The idea that the Atman is eternal, and becomes reborn over and over again is central to the concept of reincarnation that is taught in the Upanishads. This concept of rebirth is highly tied to the teachings of Karma: the future consequences of one's current intentions, thoughts, behaviors and actions. It is the accumulation of Karma that binds us to Samsara, the cycle of death and rebirth. To escape the endless cycle of Samsara requires one to attain enlightenment through the realization of Atman/Brahman. It is this state of Self realization that the majority of the Upanishads attempt to describe and encourage us to achieve through the yoga practices of meditation, mental discrimination and mantra recitation.
These spiritual concepts have exerted a profound influence on the development of Yoga, Hindu and Indian philosophy. While the yogic practices taught in the Upanishads were primarily mediation based, these philosophical teachings will remain the core beliefs for all of the future developments in the many paths and practices of yoga.
Reccomended reading: The Upanishads by Eknath Easwaran