Chemoheterotrophic, multicellular organisms that possess an actively motile and multicellular form.

There actually are quite a number of ways to define or be an animal, but multicellular, at some point motile (other than just as or zoospores), and chemoheterotrophic are sufficient to distinguish them from protozoa, fungi, and plants, respectively (and also algae).

We can also consider animals in terms of their common :

The animals are distinguished into numerous phyla, which is the major (i.e., domain, kingdom, phylum, class, order, family, genus, species). These include, for example,

phylum Porifera (the sponges)
phylum Cnidaria (the , etc.)
phylum Platyhelminthes (the flatworms)
phylum Nematoda (the roundworms)
phylum Annelida (the segmented worms)
phylum Mollusca (the mollusks)
phylum Arthropoda (the insects, etc.)
phylym Echinodermata (the , etc.)
phylum Chordata (the vertebrates, etc.)

Video (Evolution of animals, very nicely done; "Evolution 2/7 Great Transformations")