Here is a pictorial representation of a family tree of different types of grain.
First, notice the group triticeae that consists of wheat, barley, and rye. Spelt is not on this picture but is also in this group (as far as genetics goes, spelt a kind of wheat). These are the gains that contain gluten. They are also the grains that all rabbis agree have hametz.
Oats are almost as closely related (group pooideae). Some people with celiac also cannot eat oats. Most rabbis consider oats to have hametz.
(In the U.S. almost all oats are shipped in the same train cars as wheat. So almost all oats, oatmeal, and food that contains these have gluten contamination even if listed as "no gluten ingredients". A few companies do sell genuinely gluten-free oats or oatmeal shipped specially and tested for gluten after processing.)
Not closely related to wheat are rice, corn, millet, and sorghum. These grains are not a problem for people with celiac. Nearly all rabbis consider these grains to not have hametz. Amaranth and quinoa are not pictured but also also in this final category of grains distant from wheat.
