I take this question not to mean "what is the existing classification", but rather, what is the process we go through to create that classification. This question is worth a full book to answer - so clearly my short answer will be simplified & leave a lot out.

All animals (which I will here take to mean multi-cellular heterotrophs)share a common ancestor about 700 million years ago. That organism probably looked like a completely insignificant little "worm" a millimeter or so in size, and with characters far more simple than those of most living "worms". The descendants of that initial "animal" quietly diversified until about 575 million years ago when environmental changes allowed the descendants to start to explore different ways of making a living (= catching food). Some stuck up in the water column & filtered.

Some burrowed horizontally (and later vertically) on the ocean bottom. In time several learned to swim, and from both crawling and swimming forms, predators evolved. OK, what is described is an "ecological explosion" - a diversification based upon ways of feeding. Each solution to feeding was associated with certain characters - bivalves developed shells and necks, arthropods developed exoskeletons and legs, chordates developed a notochord and took off into the water column to swim. These characters distinguished all subsequent descendants of these initial "solutions to life", and thus we recognize arthropods by their exoskeleton and repeated body segments, and chordates by their notochord and then backbone. These become the "great groups" of animals.

Within each group a similar process played itself out, with certain descendants specializing in different ways of making a living. Within chordates, for example, two groups have radiated on land - reptiles and mammals. Within the mammals, some are herbivores, some carnivores, some fly and some have even gone back to the ocean (whales and friends). Each group has evolved characters that suit it to its mode of life. While all mammals are clearly chordates, within mammals, we would distinguish each group (classify it) on the basis of the characters it possesses - bats with wings, whales with loss of hind limbs, etc. And within each of these groups we would further distinguish them - not on what they have that makes them similar to other mammals, but by what separates them.

These latter characters (those distinguishing the smallest units - the genera and species) are often VERY small, and require careful attention to perceive.

In sum, we end up classifying animals using two sorts of data. We recognize the "great groups" on the basis of the characters that all their members have in common because they all descended from a common ancestor. But we distinguish between these groups on the basis of the characters that separate them - that one group has and another lacks. Within each group we continue the process, seeking to erect smaller "great groups" (mammals and reptiles within chordates) again on shared similarities from the common ancestor of each group, even as we simultaneously distinguish between mammals and reptiles on the basis of their unique characters.

So, I hope I (a) provided an answer, and (b) to the question asked.

Life forms are traditionally classified based on a system devised by Carl (also spelt Carolus) Linnaeas in 1759, which consists of a ranked scheme in which an individual species forms the lowest rank, and species are grouped into successively larger groups. The ranks are, from largest to smallest, Kingdom, Phylum, Class, Order, Family, Genus, Species. All animals belong to the Animal Kingdom. Other kingdoms are for plants, fungi, bacteria, etc.

Charles Darwin (Origin of Species, very last page of chapter 14, published in 1859) said that we classify lifeforms into these groups based on what they are related to. Animals, for instance, resemble one-another more than they resemble plants because animals had an ancestor in the less distant past than any animals have with plants. The methods for figuring out who is related to whom are still in dispute.

Traditionally, scientists have just looked at the animal or plant in question, or looked at how it develops as an embryo, and compared it to other known animals and plants. Since then, biologists have taken to trying to looking at the genes of different life forms to see who is related to whom. Rarely, you can actually see two groups split in the fossil record, telling you not only how closely they are related but how long ago they split. The fossil record is ideal for this purpose when the groups breaking apart are fossilized, but this is rarely the case, so fossils cannot be used in all (or even most) cases. As for physical form versus genes, both have their problems, so it's not as simple as saying which is better.

Scientists are currently asking themselves about your question: how should animals be classified?

Scientists used to classify animals based on the similarities of their structures, and similar animals that couldn't breed with each other were given names as different species. Now we have a lot more information, especially about the DNA sequences in the genes of the different animals, and we are changing lots of the classifications.

The scientific name of an animal now is in Latin and consists of the Genus (Homo, for us) and the species (sapiens, for us). Some scientists want to give every animal - and plant and microscopic organism - a different number and arrange them all into 'clades' instead of into orders and families and the other divisions we use today. It is all very complicated now. We used to think everything evolved directly from one animal to another, but now we're finding that the evolution is more complex, because sometimes a particular type of animal will have a gene that got transferred to it from some distantly related animal - that's called 'horizontal gene transfer'. As you can see, now it's harder to draw the family tree for all the animals.

Check out more about this on the link below:
the animal kindom