Thank you for your interesting question. Many of my first-semester biology students are taken aback when they first hear me refer to fish, birds, and insects as animals (fauna). The fact is that there seems to be a stream of belief that only mammals–lions and tigers and bears and the like–are animals. The animal kingdom, however, is vast and varied.
There are nine major animal phyla. In order from simplest to most complex, then, they are the:
1. Porifera or the sponges;
2. Cnidaria which includes the jellyfish and corals;
3. Platyhelminthes, or the flatworms, includes tapeworms and flukes;
4. Nemotoda or the roundworms;
5. Annelida, or the segmented worms, includes earthworms and leeches;
6. Mullusca includes the terrestrial snails as well as marine bivalves, squids, and oysters;
7. Arthropoda, or the arthropods, are animals with external skeletons (or exoskeletons) and include both terrestrial and aquatic and marine forms including insects, spiders, crabs, and lobsters.
All of the aforementioned phyla include animals without backbones. In other words, invertebrates. So to answer one part of your question: yes, invertebrates (which includes insects) are animals.
The final two animal phyla are the vertebrates (or animals with endoskeletons). They are the:
8. Echinodermata, or echinoderms, are animals with “spiny” skin and include sea stars (or “starfish”), sand dollars, and sea urchins;
9. Chordates include the most complex animal forms; they have endoskeletons that protect internal organs while providing shape and structure, backbones that surround and protect spinal cords, and skulls that surround and protect brains; they have complete digestive systems, closed circulatory systems, and tails at some point in their development.
There are six classes of vertebrate chordates including the:
1. Chondrichthyes, or fish with skeletons made of cartilage like the sharks and stingrays;
2. Osteichthyes, or fish with skeletons made of bone, includes salmon, perch, trout, sea bass, catfish, etc;
3. Amphibians are semiaquatic vertebrates and include frogs, toads, and salamanders;
4. Reptiles are vertebrates with scaly skin and include snakes, lizards, and turtles;
5. Aves are the birds;
6. Mammalia are the mammals and include animals with hair and mammary glands; the mammals include those lions and tigers and bears mentioned in the very beginning. And oh yeah, humans are mammals as well.