The common list of forbidden grains is: wheat, rye, barley and oats.
Though corn (maize) is one of those grains that many people -- not just celiacs -- may be allergic to, it is not a grain that is thought to cause damage to the villi in celiacs. It is tolerated by most celiacs.
Of the common grains, rice is the favorite as it rarely troubles anyone.
Aside from corn and rice, there is a wide variety of other grains that are used in gluten-free cooking. We even use beans and peas (legumes, pulses).
The following can be milled into flour: amaranth*, buckwheat* (or kasha), chickpeas (garbanzos), Jobs tears (Hato Mugi, Junos Tears, River Grain), lentils, millet*, peas, quinoa*, ragi, sorghum, soy, tapioca, teff*, and wild rice. Many of these flours are available in health food stores. Some (like rice flour) may be available in grocery stores. (The products marked with an "*" are listed as grains to avoid by some physicians and celiac societies. See the discussion below about anecdotal evidence and possible contamination of flours for more information.)
To improve the texture of gluten-free baked goods, most cooks use one or more of the following: xanthan gum, guar gum (though this sometimes has a laxative effect), methylcellulose, or a new product called Clear Gel. These can be obtained either through health food stores, specialty cooks stores, or some of the mail order sources listed below.
Oils popular in cooking include: corn, peanut, olive, rapeseed (canola), safflower, soy, and sunflower.