I have a lot of cookbooks on my shelves, and right now these are my favorites:
The Moosewood cookbooks are classics. The older ones are pretty heavy on the dairy, but the newer ones have much less. The one I use most is Moosewood Restaurant Cooks At Home. In my opinion it has the most family-friendly recipes of all the Moosewoods. It has a lot of vegan recipes.
The Voluptuous Vegan is my absolute favorite vegan cookbook, but it is gourmet, not simple easy stuff. The paella recipe is to die for, but more work than I'm willing to put in for a regular weeknight dinner.
I can't stand the name, but How It All Vegan, has very good recipes that are relatively simple, and many of them are kid-pleasing.
I have a pressure cooker and use Vegetarian Cooking Under Pressure a lot. If you have a pressure cooker, that book is practically a necessity! This cookbook doesn't call itself vegan, but it just happens to be egg and dairy free.
Two cookbooks I've been using a lot lately are the Kripalu Cookbook, which has a lot of kidpleasing recipes (the fried tofu is divine), and the 5-Ingredient-Vegetarian-Gourmet, for ideas for when I have to make dinner in a big hurry. Neither of these are vegan, but they have plenty of nondairy recipes.
Vegetarian Planet is another great one, but it has a fair amount of dairy.
I can't believe I forgot one of the best -- Feeding the Whole Family, by Cynthia Lair. Great for families, of course.
If you want a healthy vegan dessert cookbook, try Great Good Desserts.