Gluten is a protein found in wheat, barley, and rye. A lot of people have a hard time digesting it. Wheat is the basis of white flour, which is used to make bread and pasta. (Yes, some people, surprisingly, don't know this. Note that wheat-free does not mean gluten-free, but gluten-free does mean wheat-free. Also, rice, potatoes, and corn do not contain gluten.) At restaurants that use prepackaged items, gluten can be hidden in soup bases, condiments, and salad dressings in the form of modified food starch or wheat flour.
Certain Asian restaurants (mostly Chinese, Korean, and Japanese) can be a landmine of gluten because most of the sauces contain wheat flour as a base — including, but not limited to, soy sauce, oyster sauce, hoisin sauce, ponzu sauce, and gochujang (sweet and spicy red pepper paste found at Korean restaurants). Also, many non-wheat-based noodles (even ones that are made of potato, mung beans, or buckwheat, which are technically gluten-free in pure form) almost all have some wheat flour added to them (if they're store-bought). Rice noodles at Vietnamese and Thai restaurants are usually 100-percent rice-based, but that still doesn't mean they're safe (see below).
Many health foods also harbor gluten. Vegan substitutes for meat are often made of wheat gluten. And spelt is off limits, too.
Watch out for gluten in liquid form, as well, such as beer (sometimes an ingredient in salsa or marinades), malt vinegar (an ingredient in Worcestershire sauce), and barley tea (served at Korean restaurants). In other words, gluten can be hidden anywhere, in any type of food, in any amount.
Even if an item isn't made of gluten, it can still be contaminated with gluten via pans, utensils, preparation space, grills, toasters, and ovens. That's why French fries are usually off the table for celiacs: Restaurants usually have one deep-fryer, so if it serves something that's breaded and deep-fried, chances are the fries have gluten on them. Similarly, if you're at an Asian or Italian restaurant, keep in mind that pasta and noodles are often boiled in the same water. So ask that your gluten-free pasta or noodles be boiled in separate water. At one Vietnamese restaurant in San Francisco, I was told that even the vegetables in the soup are boiled in glutinous-noodle water, so the only truly safe thing was a salad.
Speaking of salads, always ask to hold the croutons (removing them from an already-crouton-tossed salad is not okay). Is that fish grilled on the same grill as something gluten-y? Depending on how sensitive you are, that could be a problem.
Even foods that are technically gluten-free, such as oats and grits, are often processed in the same facilities as wheat and thus should be assumed to contain gluten, unless specifically packaged as gluten-free. Other technically gluten-free foods include quinoa, amaranth, millet, teff, buckwheat, risotto (rice), and polenta (corn), but as mentioned, these could be contaminated either by preparation, cooking method, or additives. Always double-check to be safe.