Terraforming refers to the process of manipulating the atmosphere, surface and temperature of planets, moons, asteroids, or any other celestial body, in order to make it more like Old Earth (Terra) and supportive of human and animal life.
As humanity left Old Earth during the Hegira, it encountered many worlds where life would be possible but that needed some adaptations, to varying degrees: from small adjustments to full-scale planetary manipulations. Mars was naturally one of the first worlds to be terraformed, and by the time of the Web Era, it had a breathable atmosphere, forests, running water, and so forth.
At least at first, little weight was given to native ecological concerns when decisions to terraform worlds were made. Garden was a planet that had a complex alien biosphere, complete with sentient beings (the Marsh Centaurs), but that was completely terraformed, leading to the extinction of all native life. The Consul participated in the process of transforming Garden.
Another world that underwent advanced terraforming was Sol Draconi Septem, an frozen world that was terraformed to the point that the regions around the equator were habitable and filled with fauna and flora. However, pressure from the All Thing prevented the heating up of the polar regions, preventing the extinction of the native alien lifeforms known as Wraith.
Heaven's Gate was another initially utterly inhabitable noxious world that was turned into a lovely planet. In sum, it seems that mostly every planet mentioned in the Cantos underwent some level of terraforming. It seems that the planet of Hyperion was naturally habitable, as its natural ecosystem is left in place (see Flame Forest, the Radiant Gossamers, etc.), though Old Earth lifeforms are present such as ducks in the extensive fens on the world in which Raul Endymion was a hunting guide.
Some terraforming depended on the use of Containment Fields, such as the one holding in Tau Ceti Center's atmosphere. Some planets seem to have been habitable only in some places through the use of containment field domes, such as in Freeholm. Castel Gandolfo is a terraformed asteroid belonging to the Catholic Church - its atmosphere was similarly held in by containment fields.