It sounds like you have either atopic dermatitis (eczema
) or a contact dermatitis.
Contact dermatitis is inflammation of the skin that is provoked specifically by contact with certain irritants. Common offending agents include household cleaning chemicals. You can prevent this by wearing gloves whenever performing chores, and especially wearing waterproof gloves when washing dishes. Keeping the skin well moisturized with a good moisturizing product is a good idea as well.
Atopic dermatitis is a lot like contact dermatitis (causing rash, itching, and cracking of the skin) except that it is not so clearly associate with coming in contact with a specific agent. Definitely having the hands submerged in water such as when washing dishes will make it worse, so you should avoid that. The mainstay of treatment is skin moisturizing, and often it is necessary to apply the moisturizer many times throughout the day, especially after each time you wash your hands. You can also apply an over the counter topical steroid cream to particularly stubborn areas of redness or itching.
Talk to your primary care doctor
as soon as possible, as they will be able to confirm which condition you have and will also help you develop a good plan for addressing it.