Egg Matza is usually made with oil and/or eggs and/or sugar, using fruit juice instead of water to prepare the dough. The Rabbis in the Talmud stated that liquid-food-extracts (juices, oils) do not cause flour to leaven like water does. Egg Matza then is not made necessarily with actual eggs, but mainly with no water (real or regular Matza = flour + water). When certified Kosher for Pesach, egg Matza can be consumed during the eight days of Pesach and even during Pesach eve (when is customary to refrain from eating real Matza), but is not suitable for fulfilling the Mitzva of 'eating Matza' in the Pesach Seder itself.
In America Egg Matza is usually sweeter than regular Matza and it is elaborated in the shape of Matzot, which sometime might create a confusion: some people might think that those Matzot can be used for the Seder, when technically speaking, they are not consider Matzot, but non-chametz cookies...
Among some Ashkenazi Jews, there is a custom not to eat egg Matza during Passover at all, except for the elderly, infirm, or children, who cannot digest plain Matza.
Among Sephardic Jews, this type of Matza or baked foods--when the dough is prepared with fruit juices or oil and does no contain any amount of water--is called Matza 'ashira or enriched Matza,and unlike egg Matza, products under the category of Matza 'ashira do not necessarily have the shape of regular Matza. Rather, as you might verify in your local Kosher grocery, Matza 'ashira products come in the form of cookies, cakes, desserts, etc. which actually look very similar to chametz products. But as we said, since no water was used in their elaboration, if it bears the corresponding Kasher le-Pesach certification is OK for Sephardim to use on Pesach.