Like all true grains, rye is a type of grass and is closely related to wheat and barely. Do not confuse rye with ryegrass, a type of grass used for lawns and feeding animals.
It is important to distinguish between celiac disease or gluten allergy vs. wheat allergy. If you are allergic to gluten or have celiac disease, you must also avoid rye. However if you have a wheat allergy, you may be able to tolerate rye.
Note that rye can become infected with the ergot fungi, which is the original source of LSD. This is no longer common due to mondern food handling programs.
Benefits of Rye
For those of you who gleefully consume gluten, rye does have health benefits. Most of these are related to whole grain rye which is high in fiber; the same is true of any whole grain. There are reports (see reference  below) that claim rye lignan (lignans are a class of chemical compounds produced by plants) helps with menopause as it acts in a similar manor to estrogens.
 Wikipedia on Rye