My hands and feet have the red scaly itchy peeling skin. Looks like the palm on the web site. What is that?

Fungus. Is the most likely cause. But many things can cause this. I suggest you see a dermatologist.
Eczema. Sounds like it may be dishidrotic eczema. Could be contact dermatitis from some exposure to chemicals or too frequent hand washing or it could be a fungal infection. I would see your doctor. In many cases avoidance, lotion/emollients, steroid cream and/or antifungals are needed. It should be treatable though and can be a very troublesome condition.
Psoriasis. Could be either psoriasis or a fungal infection. Go see a dermatologist or a podiatrist so that a biopsy of the skin can be performed to find out exactly what it could. I have also see scalded skin syndrome that appears as peeling itchy feet. This is bacterial infection.