Updated: May 2
Although most vegetables need at least 6 hours of sunlight to grow, there are a few that will grow with as little as 3 to 4 hours of sunlight throughout the day. If you have limited sun on your balcony or in your backyard, don't despair, there are still a variety of herbs and vegetables that you can grow.
When I lived in town and had a shady, east facing backyard I personally had success with rhubarb, strawberries, green onions and mint. My lettuce ended up bitter, but I didn't know better and had planted a head lettuce. Green onions had the best success when I left them in the ground over the winter. That way they had a head start and were ready for harvest early in the spring.
Plants for food that can be grown in shady areas of just 3 to 4 hours of sunlight include:
lettuce (avoid head lettuce for shady areas but instead choose a loose leave variety).
strawberries (Many say strawberries need 8 hours of sunlight but I found mine did well in my part sun location.)
If you have as many as 5 to 6 hours of sunlight in your garden area you can grow most any garden vegetable, although many of them may be smaller than they would otherwise be in a sunnier location. Beets, peas, beans, broccoli and cabbage are popular vegetables that can grow well with around six hours of sunlight. Keep in mind they will do much better if the soil is warm. This may mean a raised bed, container or a pot as the soil in these will be warmer than ground soil. Experiment and see what works best for you.
Plants that thrive in full sun locations (minimum 8 hours) include most root vegetables (with the exception of beets) corn, and most plants that produce "fruit", such as squash, cucumbers, tomatoes and peppers.