Definition - What does Kaivalya mean?
Kaivalya is a state of solitude, aloneness, isolation and detachment. The word is derived from the Sanskrit kevala, meaning "alone" or "isolated." It is a separation of purusha (Self or Soul) from prakriti (primal matter). The state of kaivalya is the main goal of Raja yoga. It is a detachment and independence from relationships, egoism, attraction, aversion and the cycle of birth and death. One can achieve this state by performing austerities, yoga practice and discipline. One who achieves this state is called a Kevalin.
"The Yoga Sutras of Patanjali" talks about a yogi who achieved kaivalya and is independent from all bonds. He attained the state of absolute consciousness, described in the chapter titled "Samadhi Pada."
Yogapedia explains Kaivalya
According to Raja yoga, kaivalya is the final stage of enlightenment that a yogi can reach, also called moksha or nirvana. In this state, the yogi becomes completely fearless and free. Although it is sometimes wrongly perceived as negation or annihilation, kaivalya is a state of total awareness.
A yogi must isolate his/her soul completely from the matter and understand that his/her happiness is not connected to the external word. He/she realizes his/her independence and importance and then achieves kaivalya.
Someone who wishes to attain kaivalya must overcome his/her desires and attachments. A kevalin is free from the modifications of the mind as well. As Patanjali says in the Sutras, "the soul finds its end in freedom, enlightenment and kaivalya."