Animisim is a kind of religion in which people believe that various non-human things (whether living or not) have spirits. Animists believe that these spirits have an impact on what happens in the world.
Most animistic religions are associated with what we would call "primitive" societies. These are societies which lived in close and constant contact with nature and who therefore felt that natural things had an impact on their lives. These would be people who believed in the spirits of rocks and trees and wild animals and would conduct rituals to appease or propitiate these spirits.
There are "modern" religions with animist aspects. A major one of these is the Shinto religion of Japan. True believers of Shinto believe that many natural things have spirits known as kami. Believers would do rituals to honor the kami and to invoke their aid.