What are the different types of eczema?

Too many. There are too many to discuss but seborrheic , atopic, nummular, and contact are the common ones.
Whole variety. Eczema is primarily a morphologic term used to describe a red, scaling, inflammatory eruption with itching, edema, papules, vesicles, and crusts. Included are: atopic dermatitis, allergic contact dermatitis, nummular eczema, wiscott-aldrich syndrome, seborrheic dermatitis, juvenile plantar dermatosis, etc.