What everybody said above is pretty much right on. I will just add that eczema is what they call a multi-symptom disease and that the basic underlying cause, at the cellular level as it were, is dry skin. Therefore the cure is to avoid all irritation of the skin and keep moisture locked in to the skin so that it can function as the protective barrier it was meant to be.
This is done by avoiding soap, keeping clean (sounds paradoxical, I know), and using moisturizers like Cetaphil.
Eczema is actually NOT an allergy although it is often seen with allergies. Any sort of irritation or upset can start the eczema. So therefore an allergic reaction can set off eczema, or stress can set of eczema, or an abrasion to the skin can set off eczema. It's actually pretty interesting how it all works together.
Once the eczema gets going, it can develop from a rash into infected skin which as I understand it is doubly hard to treat because you have the initial rash/delicate skin condition plus a bacterial or fungal infection. This is where the really bad inflamed skin comes from.
Sun seems to be good for eczematic skin, although of course you don't want to get too much. The reason why is not fully understood yet.
My lo had mild eczema that is almost completely cleared up at this point. We treated her red seepy areas with corn starch powder when she was very young, which dried it out enough to apply lotion (again, sounds paradoxical). Now the only treatment she needs is using a none-soap cleanser, using a gentle detergent on her clothes and sheets, and applying Cetaphil lotion after her daily bath or swimming.
Other precautions I take--having her wear jeans outside so she won't scratch her knees, and quickly disinfecting any abrasions. I also take care to avoid heat and stress, both which can cause an outbreak for her.
Some people notice an eczematic outbreak in response to some foods. This is not always an allergy! Sometimes if the introduction of a food is stressful to the digestive system, eczema is the result. But when the digestive system adjusts (within days), the eczema stops. This was the case with Luka. She had no food allergies, but she would have stress outbreaks. Then she'd adjust to the food and be fine.
If you want to read more about eczema in a scientific vein, the New Zealand dermatological society has the best site I've ever seen.
http://www.dermnetnz.org/Good luck and I hope your lo has a full recovery from her symptoms!