Naturally I had to add my two cents worth...
Leeks - only use the white and pale green parts. They're hardier than other onions, so they work best in soups & stews. Cleaning tips from a Jamie Oliver episode I saw...
Slice through the leek lengthwise starting just below the hairy roots. You don't want the leek to completely fall apart before rinsing.
Carefully rinse the sand from them, holding the hairy part closest to the tap so that the sand runs down and doesn't accumulate in the delicious white part.
Red onions are usually sweet and can be cooked or eaten raw in salads or sliced. I love them topping a burger or with bagels, lox & cream cheese.
Yellow onions (the ones most commonly found in supermarkets) are the most versatile. And they make the best caramelized onions that are so important, especially in my favorite French Onion Soup. They are one part of French, Italian and New Orleans "trilogy" - along with carrots & celery to flavor roasts, stews.
Green onions or scallions or Spring onions are very mild and can be eaten raw - I love them thinly sliced or chopped and sprinkled over Asian dumpling soups. I even love them quickly grilled on the barbecue to accompany some grilled chicken or beef.
Shallots are my newest love of the onion family. I finely chop them to make awesome vinaigrettes and caramelize them for so many dishes. When cooked slowly they are as sweet as candy.
Chives - almost garlic-tasting, and when finely chopped are the perfect addition to great mashed potatoes.
If you have any other suggestions, please leave a comment.