Contrary to popular belief in the west, the aim of yoga is not intended to increase flexibility, or sculpt the shape of your rear end.
The word yoga stems from sanskrit work 'yuj' which means "to unite" or "bring into union".The ultimate goal of yoga is 'moksha' or "liberation" from suffering.
The various disciplines of yoga are practices that can be used as tools to enter into and connect with supreme consciousness. By supreme consciousness I mean the consciousness which permeates all things (for example, each seed has an innate wisdom which tells it to send roots into the earth and stem and leaves towards the sun.) We enter this all encompassing consciousness through deep awareness of the self.
In the most simple terms, in Hatha Yoga we enter through the channel of the physical body, becoming deeply mindful of our bodies as we move through physical asanas. In pranayama, we practice bringing awareness into the prana body through mindful and intentional breath. In raja yoga we practice awareness of the mind. Bhakti yoga is the practice of conscious loving. Karma yoga, for example has a goal of transforming all of our actions into selfless actions.
Each yogic practice possesses within it, the capacity for deep and radical transformation. The practices often lead from one into the next, beginning with the most simple, which is to become deeply aware of our physical body and being-ness. As we move deeper and deeper into these practices we unite our bodies, minds, and spirits to create more whole beings within our selves.
The great yogic sage Sri Aurobindo explains it as this:
"The aim of the yoga is to open the consciousness to the Divine and to live in the inner consciousness more and more while acting from it on the external life, to bring the inmost psychic into the front and by the power of the psychic to purify and change the being so that it may become ready for transformation and be in union with the Divine Knowledge, Will and Love. Secondly, to develop the yogic consciousness, i.e., to universalise the being in all the planes, become aware of the cosmic being and cosmic forces and be in union with the Divine on all the planes up to the overmind. Thirdly, to come into contact with the transcendent Divine beyond the overmind through the supramental consciousness, supramentalise the consciousness and the nature and make oneself an instrument for the realisation of the dynamic Divine Truth and its transforming descent into the earth-nature."
-- Letters on Yoga, SABCL Vol. 22/24, p. 509