I know that plasmids are circular, and so they can be found in archaea and bacteria, as these also have circular genomes. But what about plants and animals? Do they naturally have plasmids too? and how is the mechanism of action when plasmids are used to change the phenotype of plants and animals?

The closest thing in eukaryotes is mitochondrial DNA which is thought to have evolved from bacterial plasmids. See http://en.wikipedia.org/wiki/Mitochondrial_DNA.

As you say plasmids are used routinely in cell biology research on mammalian cells to add in new genes to the cell which then allows their function to be studied. That is however done in a controlled environment in a lab and only very rarely in humans and even then under very strict controls and guidelines.