Barley and wheat are both cereal grains, but barley is not a type of wheat. Einkorn, emmer, spelt, durum and bread wheats are the major types of wheat. Barley is in a different genus (_Hordeum_) than wheat (_Triticum_).
[Edit: But, as QuercusMax mentions, they are in the same Family (and Subfamily) along with rye and oats. Corn and rice are also in the same Family, but not the same Subfamily, as wheat and barley.]