The term involution refers to different things depending on the writer. In some instances it refers to a process that occurs prior to evolution and gives rise to the cosmos, in others an aspect of evolution, and still others a process that follows the completion of evolution in the human form.
Involution according to Esoteric cosmology
In Theosophy, Anthroposophy and Rosicrucianism, involution and evolution are part of a complex sequence of cosmic cycles, called Round. When the universe attains a stage of sufficient density, the individual spirit is able to descend and participate in the evolution. Involution thus refers to the incarnation of spirit in an already established matter, the necessary prerequiste of evolution:
- As an example, the so-called descent of the Monad into matter means an involution or involving or infolding of spiritual potencies into material vehicles which coincidentally and contemporaneously, through the compelling urge of the infolding energies, unfold their own latent capacities, unwrap them, roll them forth; and this is the evolution of matter. – Gottfried de Purucker 
That period of time devoted to the attainment of self-consciousness and the building of the vehicles through which the spirit in man manifests, is called involution. Its purpose is to slowly carry life lower and deeper into denser and denser matter for the building of forms, till the nadir of materiality is reached. From that point, life begins to ascend into higher Worlds. This succeeding period of existence, during which the individual human being develops self-consciousness into divine omniscience, is called "spiritual evolution."
Involution according to Sri Aurobindo
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Sat, Chit-Tapas, and Delight/Ananda are the three aspects of Satchitananda, and are mentioned in terms of Involution . Spirit or consciousness manifests as these three, and then as the intermediate link of Supermind, which is transitional between the higher and lower (matter, life, and mind) nature.
The reason for involution is Delight - the Delight of Being (the Spirit or Absolute) moving to Delight of Becoming (temporal existence, the cosmos). Being throws itself forward into a multiplicity of forms, becoming lost in the inconscience of matter, and then through evolution it partakes in the Delight of rediscovering the Spirit which had been hidden in the interim.
Evolution is thus the movement forward by which the created universe evolves from its initial state of inconscience (i.e. as matter), evolves animated life forms and mental beings (i.e. humans), and continues to evolve spiritual properties, and in that process rediscovers its Source. Such an Evolution of animated forms is only possible because at each stage of development, the developing entity contains within itself the conception of what it may become. Thus, the evolution of animated life out of matter supposes a previous involution of that animated capacity. This is akin to a seed that already has the essence of the tree that will emerge from it.
Each plane emerges from an earlier plane through the evolutionary process, which takes place in chronological time. But in a parallel construction, each of these new planes can be understood as being a descendent of its corresponding higher order plane from the Infinite. Thus, when mentality emerged in the universe, the universal plane of Mind was implanted to a degree in those beings harboring that mentality.
The evolution is the development of all entities in the cosmos, including humans, in order to attain their fulfillment, including the discovery of spiritual Delight, which was, and always is, the experience of the Source Creator. The evolution is the progressive development from the original inconscience of matter into life (movement, sensation, desire, etc. and living physical beings), and from thence to mind (in conscious animals and most especially humans - the self-conscious thinking animal), and from thence to spiritualized mind, culminating in the The Supermind or Truth Consciousness (as supramental individuals, and finally the supramental, i.e. a divine life on earth.) 
|This section requires expansion. (June 2008)|
We cannot speak of Sat without Chit Ananda or Being, Consciousness and Bliss. They represent a totality. Sat is the vital state of that which is was and always will be. In a sense it is a beginnining but because it is pregnant with possibility it is inexorably tied to Ananda or the recognition of Being and then the subsequent realization of bliss which is divine inner knowing. Each flows out of the other and then back again. It could be said that Sat only exists through Ananda or Consciousness however, these levels of differentiation cannot grasp the true nature of either of these three qualities since they are interdependent.
Chit-Tapas or Consciousness-Force, in Sri Aurobindo's philosophy refers to the pure energy of Consciousness by which creation ultimately comes about; the infinite divine self-awareness which is also the infinite all-effective Will. It is also one of the seven planes of existence, according to the Vedic cosmology and the seven lokas of Hindu thought.
In chapter 10 of The Life Divine, Sri Aurobindo writes at length on the nature of Consciousness-Force as a principle of the Divine. As he understands it (following the Tantric dichotomy of Shiva and Shakti), Chit or Consciousness is not an inert and passive principle; but contains the potential spiritual Energy, Tapas, which in Creation becomes the dynamic and creative principle or Force, called Shakti. Chit-Tapas or Chit-Shakti is therefore the universal Consciousness-Force, the divine Energy; the Mother.
Delight is Sri Aurobindo's term for ananda, and plays a large part in his cosmology and spiritual teaching. Delight is the reason for creation, by which The Absolute extends its Delight of Being into multiplicity, losing itself in the inconscience and then through Delight rediscovering Itself through individuals realising their Divine nature and proceeding to spiritual realisation.
In other words, the universe was created so that the Delight of the Infinite Spirit can manifest in all the forms of creation. When we discover our higher nature, the soul and spirit, we experience the delight for which we were came into being and of which we are a part.
In chapters 11 and 12 of The Life Divine, Sri Aurobindo writes at length on the nature of Delight as a principle of the Divine, and its role in creation.
Meher Baba uses the term "involution" to mean the inner path of a spiritual aspirant toward Self-realization. He divides involution into seven stages he calls "planes," and describes different experiences and powers had on each, until the Goal of full enlightenment is achieved at the seventh plane.
Other Indian interpretations
For Baba Hari Dass, Evolution is the Cause of Involution, and vice versa. (Baba Hari Dass is a Maunisadhu monk who practices continual silence.)
Swami Sanathanananda Puri uses the terms evolution and involution in a spiritual context to elucidate the PariNama VipariNamana Siddhanta or the Cyclic theory of Evolution and Involution. Through intense tapasya and exhaustive research, Swami Sanathanananda Puri has restored this theory of the ancient Rishis. The Supreme Being has effectuated creation through ParaMAnu (परमाणु) or primordial atoms. Each ParamANu follows a cyclic path of descent and ascent. Evolution or PariNama denotes the first half of the cycle, while Involution or VipariNamana denotes the concluding half. Evolution is the process where a ParamANu descends from the realm of Atma (Atma Thal) to the realm of Jiva (Jiva Thal). Involution is the ParamANu's return journey or ascension from Jiva realm to Atma realm. While undergoing evolution, the ParamANu successively assumes and discards a number of bodies of those of plants and animals. It finally attains a human body upon completing evolution. Then starts the return journey or involution process, which is termed as VipariNamana. VipariNamana is the ascension of ParamANu to its original state in the Atma realm.
In integral thought, involution is the process by which the Divine manifests the cosmos. The process by which the creation rises to higher states and states of consciousness is the evolution. Involution prepares the universe for the Big Bang; evolution continues from that point forward. The term involution comes from the idea that the divine involves itself in creation. After the creation, the Divine (i.e. the Absolute, Brahman, God) is both the One (the Creator) and the Many[disambiguation needed] (that which was created).
The integral philosopher Ken Wilber refers to involution in his online chapter of Kosmic Karma, employing concepts from Plotinus, Advaita Vedanta, Tibetan Buddhism, and Sri Aurobindo. According to Wilber, the cosmic evolution described in his previous works is preceded by an involution of Spirit into Matter. This involution follows the reverse stages to the sequence of evolution - e.g. Spirit to soul to mind to life to matter. Once the stage of insentient, lifeless matter is attained, then "something like the Big Bang occurs", whereupon matter and manifest world come into concrete existence, from which stage evolution follows.
- Gottfried de Purucker, Occult Glossary, TUP, 2nd ed., 1996 
- Sri Aurobindo, The Life Divine, Sri Aurobindo Ashram, Pondicherry, 10th ed. 1977.
- The Life Divine Bk I, Ch.11-12
- The Life Divine bk II, ch.27-28.
- Wilber, Ken, Introduction: From the Great Chain of Being to Postmodernism in Three Easy Steps, Shambhala Publications, 2005.