The green stems are the part of the chives that are used as an herb. Chive stems are hollow and are usually used fresh. Chives are typically chopped and can be used as a garnish, although they do have a mild oniony flavor, especially fresh ones.
Chives can be featured in all sorts of recipes, from baked potatoes to soups, salads, sauces and omelets. They're frequently mixed with cream cheese to make a savory spread. Chive butter, a compound butter made by blending chopped fresh chives into butter, is frequently served with grilled steaks or roasted poultry.
Chives are related to but not the same as garlic chives. Garlic chives have wider, flatter stems which are not hollow, and they have a rather pronounced garlic flavor.
Chive plants develop bright purple blossoms which are edible and also have a mildly garlicky, oniony flavor.
Also see: Quiz: Is it an Herb or a Spice?