Animism is the belief that nature is not just alive, but aware - perhaps conscious, and a thing with which humans can have a personal relationship.
But animism comes in various brands.
1. The ancient pagan or hunter gatherer type animism is mostly about power - about building alliances with natural phenomena.
At least that is the basis of 'shamanism' - to use, and benefit from, contact with, and a relation with, the spirit world of nature (animals, large plants such as trees, perhaps features such as a river or mountain).
Such contact being made in states of altered consciousness such as trance or sleep. The possibility of such a relation derives from the belief that 'life' circulates through the world, undergoing transformations - so boundaries between humans, and between humans and animals and other animated entities, are fluid. Life circulates as a spirit, by reincarnation, by transformation of forms - the shaman can therefore participate in relationships with other spirits.
Benefits sought might be wise advice (about hunting, finding food or water, when and where to move base, discernment of various types...), or healing, or success in some project (e.g. victory over rival tribes...).
(Something very similar to hunter gatherer animism is spontaneous to humans, and a feature of all young children - making adjustments for their more limited cognitive capacity. )
2. Christian animism sees nature as alive, conscious, purposive and intelligent - because animated by angels (and demons).
This is perhaps harder for modern people to conceptualize than is hunter gatherer animism - the best model I have seen is in Rupert Sheldrake's The Physics of Angels (and ignoring the confusing contributions of his co-author).
Sheldrake models angels by analogy with physical fields (like gravity, magnetism) and using concepts from quantum physics and relativity to understand difficult concepts such as how angels (like photons) have no mass (like fields), yet act locally (like particles).
Christian animism does not see animated nature as a potential source of powerful alliances, but as a description of reality that reveals the glory of God and the nature of His universe.
3. Modern, neo-pagan animism is primarily therapeutic: it seeks to cure the modern disease of alienation, to alleviate the sense of meaninglessness from the feeling of an isolated consciousness in a dead universe.
Modern neo-pagan animism is therefore a descendant of the psychological project of Jung. It is not about a description of extra-personal reality, but of perceived reality - and its objectivity comes from the assumption that there are universal, archetypal patterns of the human mind which may be discovered and stimulated.
So, these three perspectives share animism, but differ in the role they allocate to animism in the scheme of things.
By and large, pagans intend to use animated nature, to increase their life satisfaction whether physically or psychologically.
Christians, by contrast, revere or venerate animated nature, since the world has intelligence and of high status - but Christians also fear animated nature on earth and in time, since some of these high intelligences are malign.