Asparagus is a member of the lily family of plants. The edible part is basically the shoot of the plant, and the sooner its harvested, the more tender it is. Once the shoots get too long, or start to leaf and/or flower, in my opinion, they become too woody to eat. I'm not sure if those parts are eaten by anyone.
The most common variety of asparagus is the green variety pictured here, but white asparagus is also available, although it is more common in Europe than in the US. Interesting fact about asparagus - it was a favorite of King Louis XIV.
Nutritionally, asparagus is indeed fit for a king. Its one of the richest plant sources of Vitamin B9 (Folic Acid). It is also a great source of potassium, low on sodium and is a rich in Vitamins A, C, B1 (Thiamin) and B6 (Pyroxidine).