Short answers, pulled from the story.
Buddhist cosmology is the description of the shape and evolution of the universe found in Buddhist scriptures and commentaries. It covers a temporal cosmology, the creation and dissolution of universes across different eons, and a spatial cosmology, the vertical planes of rebirth and their horizontal groupings. The whole system is built from the five elements of Earth, Water, Fire, Air and Space, and is tied throughout to karma.
The three realms of Buddhist cosmology are the Arupyadhatu, or formless realm, with four planes; the Rupadhatu, or form realm, with sixteen planes in most counts; and the Kamadhatu, or desire realm, with eleven planes. Together the three realms hold thirty-one planes of existence.
A mahakalpa, or 'Great Eon,' is never given an exact length in years in Buddhist cosmology, only that it runs to billions of years or more. Each mahakalpa divides into four kalpas: an eon of evolution, an eon of evolution-duration, an eon of dissolution, and an eon of dissolution-duration.
A great trichiliocosm, called Trisahassi Mahasassi Lokadhatu and also known as the Galaxy, is the largest grouping described in Buddhist cosmology, made up of a billion Cakkavala world-systems. A Tathagata can send his voice and power throughout an entire great trichiliocosm by filling it with his own radiance.
Udraka Ramaputra and Alara Kalama were the second and first of the Buddha's original teachers, in Buddhist cosmology. Udraka Ramaputra reached the sphere of neither perception nor non-perception, a plane with an 84,000 Maha Kalpa lifespan. Alara Kalama reached the sphere of nothingness, at 60,000 Maha Kalpa, and both mistook their attainment for enlightenment.
Mahayana cosmology keeps the earlier framework's cyclical rebirth, its many realms, and its reliance on karma, but adds an emphasis on innumerable Buddhas, countless Buddha-fields, and universal salvation. It describes pure lands such as Sukhavati and Abhirati, and East Asian schools like Tiantai expand the traditional six realms of rebirth into a schema of ten.