In biology class, we studied a living 'thing' that defied classification as a plant or animal, can anyone tell

That can be Euglena viridis, a single celled organism with green chloroplasts (plant related structure) and also feeds and moves like a simple animal.

It is studied in advance books on Botany, and Zoology.

All life currently is classified into 5 kingdoms

The last two kingdims, Plantae(Metaphyta) and Animalia(Metazoa) are vey well known

Of the first three kingdoms, the first one is Monera and includes prokaryotic primitive single-celled organisms such as bacteria. The second kingdom is Protista and includes single-celled eukaryotic organisms. The third kingdom includes fungi which are multi-cellular saprobic organisms.

As the others mention, Eunice viridis is a protist which has chloroplasts(it can prepare its own food like plants in the presence of sunlight) and also shows mixotrophic nutrition(feeds on organic matter in the absence of sunlight).

Your Venus fly trap is a carnivorous PLANT and is grouped under the plant kingdom. It however feeds on insects occasionally by trapping them to supplement its diet with nitrogen which might be lacking in the soil. But it is very much a plant and shows all the characterisitics of being one and should not be thought of as being an animal. Other carnivorous plants include the Pitcher Plant, Bladderwort and Sundew.

I am not sure but I think you may be alluding to the Venus-Fly Trap, which is a canivorous (meat eating) plant.

Euglena. It's a protist. In the light, it photosynthesizes. In the dark, it engulfs prey.

I think you are thinking about Fungi - neither plant nor animal.

Protist.