How do you determine whether an organism is an animal?

If an organism is multicellular, heterotrophic (cannot make its own energy), and has no cell wall, it is an animal. This is the definition of the kingdom Animalia.

There are unicellular animal-like members of the kingdom Protista commonly called protozoa.

Fungi are heterotrophic and multicellular, but they have a cell wall made of chitin.