Any set of instructions on how to start a garden generally begins with "Find an area that receives at least six hours of sunlight each day", but many of us with small, urban yards can't be picky. Most edibles grow better with lots of sun, especially in cooler climates, but even if the location isn't ideal, I'd say that a reduced harvest is better than no harvest at all.
If your shady area is under a deciduous tree, there may be quite a bit of sun in early spring, before the leaves are fully out. There might be time to start quick-growing crops like radishes or greens before it gets too shady, and as the sun gets less strong in that area, these vegetables will be less likely to bolt or become bitter. This is especially true in areas with hot summers - the hotter your summer is, the more likely it is that your vegetables will appreciate a little shade.
And before the list, a rule of thumb: fruiting, hot-weather plants (like tomatoes) need more sun; cool-season leafy vegetables handle shade better. Many plants that are a little "wild" will also thrive in less ideal conditions, so don't be afraid to experiment with some new edible plants!
Tolerates light shade: Asparagus, pole beans, beets, brussels sprouts, cabbage, carrots, cauliflower, celery, garlic, kohlrabi, onions, parsnips, rutabaga, swiss chard and turnips.
Tolerates partial shade: Broccoli, garden cress, kale, leeks, lettuce, peas, potatoes and radishes. Several herbs will also do fairly well in these conditions, including chamomile, lavender, rue, sweet marjorum, cilantro, tarragon, parsley, oregano and thyme. If you have a place for shrubs or vines, you could try lowbush blueberries, saskatoon berries, currants, or hardy kiwi (Actinidia kolomikta).
Grows well in partial shade: Try greens such as spinach, sorrel, watercress, arugula, claytonia, endive, escarole, mustard, New Zealand spinach, radicchio and orach; herbs that will do well include chives, chervil, lovage and sage. If you want fruit try rhubarb, alpine strawberries, gooseberries and raspberries.
For the more adventurous partial shade gardener:
Good king Henry is a hardy, perennial vegetable from Europe usually used for spring shoots, although it also has edible leaves and flowerbuds.
Saffron crocus (Crocus sativus) is an autumn-blooming bulb whose reddish stigmas produce the extremely expensive spice of the same name.
Water celery (Oenanthe javanica) can be grown in water gardens or moist garden soil and its leaves are used in soups or salads.
Skirret is a mild-flavoured root crop that can be grown as a perennial.
Udo (Aralia cordata) is a large, hardy perennial whose shoots are used somewhat like asparagus, although they must be soaked or boiled before eating to remove an unpleasant flavour.
Okinawa spinach (Gynura crepioides) is a cooking or salad green with purple-undersided leaves that grows best during hot weather.
Scorzonera is often grown as a root crop, but its leaves can also be eaten like lettuce.
Turkish rocket (Bunias orientalis) is a perennial brassica with somewhat spicy leaves that can be eaten cooked or raw, and it also produces flowerbuds that resemble broccoli raab.
Sea kale (Crambe maratima) is a perennial kale often used for its blanched spring shoots, although it also has edible leaves and flowerbuds.
Achira (Canna edulis) is similar to ornamental cannas and can be used as a root crop.
Groundnut (Apios americana) is a hardy, perennial root crop with small tubers that are high in protein.
Water mimosa (Neptunia oleracea) is a floating aquatic plant with edible leaves.
Camass (Camassia spp.) has edible bulbs (not to be confused with death camass, Zigadenus spp.) that should be cooked in a pressure cooker at 257F for nine hours, after which it is apparently delicious.
Daylilies are usually used for their edible flowers, or the flowerbuds, which, when dried, are the "golden needles" used in Chinese cooking.
Edible hibiscus (Abelmoschus manihot) has very ornamental flowers, and its leaves can be used raw or cooked and as a thickener for sauces.
Musk mallow (Malva moschata) is another ornamental plant with similar uses to edible hibiscus.
Pokeweed (Phytolacca americana) should never be eaten once the plant shows any pink, but the young shoots are apparently tasty after they have been cooked in several changes of water, otherwise they, and every other part of the plant, are very poisonous.
Linden (Tilia spp.) is a tree whose young leaves can be eaten.
Stinging nettle (Urtica dioica) is a nutritious, hardy perennial that loses its sting with a little wet cooking.
Highbush cranberry (Viburnum trilobum) produces crops of sour berries that are high in pectin and make a decent cranberry jelly.
Vegetables for shady areas: Try horseradish, mint, or sorrel. Other options for the adventurous include:
Ostrich fern (Matteuccia struthiopteris) is used for its tender spring shoots, often called fiddleheads.
Giant Solomon's seal (Polygonatum biflorum var. commutatum) has spring shoots that are used almost exactly like asparagus.
Ramps are wild onions that resemble scallions and mostly grow in spring, going dormant in warmer weather.
Sissoo spinach (Alternanthera sissoo) forms a ground cover whose leaves are used for cooked greens.
Wood nettle (Laportaea canadensis) is a nutritious, hardy perennial vegetable said to be less invasive and to have better flavour than stinging nettle, its leaves are eaten cooked.
Bunchberries (Cornus canadensis) are a dwarf dogwood with that produce small quantities of berries. The flavour is usually described as bland, but they are high in pectin.
For more complete cultivation information on some of the odd, weedy plants listed, consult Eric Toensmeier's excellent book Perennial Vegetables. And never again think you can't grow anything to eat in the shade!