Fennel
There are three main types of fennel, all edible but each are used somewhat differently.

Bitter Fennel
Bitter fennel is native to Southern Europe and has been known since classical times. It figures prominently in the Greek myth of Prometheus, who stole fire from the gods by hiding it in a hollow stalk of fennel, thus freeing us from a legacy of brutish ignorance. Its seed is now grown mostly in Central and Eastern Europe and its popularity is limited primarily to that region. It's quite similar to celery seed in culinary applications, having a somewhat bitter flavor but the entire plant is edible, with the stalks often chopped and used in stews and pickles.
Sweet Fennel
Sweet fennel is the type best known to the rest of the world. Its country of origin is most likely Italy but it first appears in the historical record in a 9th century edict by Charlemagne in which he orders sweet fennel to be grown in the imperial gardens in what is now Southern France. Like its bitter cousin, the entire sweet fennel plant is edible and was used extensively in mediaeval cooking, primarily as a flavoring but also as a vegetable. Today, its anise flavored seed is popular around the world, particularly as a companion to fish dishes, though it is also used to flavor breads and sausages.
Sweet fennel is also used in herbal formulas and aromatherapy. It has a long history as a carminative (relieves flatulence) and digestive aid and in mediaeval times was used in dilution as an eyewash and mild diuretic.
Florence Fennel
Florence fennel, also known as finocchio, produces a fist-sized swelling of overlapping stalks at its base that is used as a vegetable. It's the most recent member of the fennel family, developed in Italy in the 17th century. It can be sliced and fried or steamed, but is commonly used raw, sliced thinly into salads, adding crisp texture and a light anise flavor.
What to Look For
Florence fennel grows best in cool weather, making it a perfect early spring crop in Northern California while an autumn or winter crop in the colder Northeast. While the entire plants of both bitter and sweet fennel are edible, they are commercially grown for their seed and are rarely, if ever, available as vegetables like Florence fennel.
Look for Florence fennel bulbs that are firm and clean. Bulbs are usually available with stalks and feathery leaves intact, so make sure the stalks are close together, not spread out, and that there are no flowers on the stalk, an indication of age. Fennel bulbs are sometimes sold with the stalks already cut off, a possible indication that they are not fresh. Avoid them if the cuts are dried and white and if there are splits or brown spots.
The Fennel seed found in the spice aisle is usually the sweet variety. It's also available in the bulk department. Look for seed that is light green in color.
Storage
Store unwashed bulbs of Florence fennel in the refrigerator crisper drawer for up to four days. Keep fennel seed tightly sealed in a dark, cool place for up to one year.
