Alliaceae: onions, garlic, chives, leeks
Apiaceae: carrots, parsley, celery, parsnip
Asteraceae: lettuce, endive
Brassicaceae: cabbage, broccoli, Brussels sprouts, turnip, radish, kale, Chinese cabbage, rutabaga
Chenopodiaceae: beets, Swiss chard, spinach
Curcurbitaceae: cucumbers, melons, squash, gourds
Fabaceae: peas, beans
Solanaceae: tomatoes, potatoes, peppers, eggplants
Plants from the same family tend to attract the same pests and diseases, and they often require similar care, so group them together. Divide the garden into at least three sections, and rotate each section to a new family each year. This will give time for pests or diseases in the soil to become less entrenched before moving plants that are most susceptible back to that location. Deep-rooted plants can also be rotated with shallow-rooted plants.
Of course, rotating your garden can be more complicated than just moving everything over each year. Tall plants such as tomatoes, pole beans, climbing peas and corn are usually planted at the north end so they don't overshadow other plants; they can't be indiscriminately switched with the cabbage because there won't be enough sun in the rest of the garden. However, simply avoiding planting vegetables from the same family in the same space for two years in a row (preferably three) should help you avoid serious pest and disease problems.