In Hindu mythology, each god or goddess rides on a special mount, known as a vahana (lit. "conveyance" in Sanskrit). The deity is usually depicted seated or standing upon the animal, although sometimes the animal pulls their chariot instead. Hindu statuary and iconography can get pretty abstract sometimes, and recognizing the mount can often be the easiest way to recognize which deity is being represented.
The vahana act as signs, servants, and messengers for the deity they serve. All animals of that type are considered sacred to that deity, and seeing one of them is often thought of as an omen that the deity is near or is watching.
The vahana are not generic animals, however - each mount is a unique quasi-deity, often with an individual name and an extensive mythology of its own. Famous vahana include Shiva's bull Nandi, Vishnu's bird Garuda, Ganesh's rat Mooshika, and Indra's white elephant Airavata. Other vahana include Skanda's peacock, Brahma's swan, Parvati's lion, Lakshmi's owl, Yama's water buffalo, and Vayu's antelope.
Kubera, the god of wealth and greed, rides on a man.