Gluten is a very sticky set of proteins. They give baked goods such as bread and pastries their strength and consistency. They do not dissolve in water, unlike the starches with which they are found.
Gluten is not so much a thing as a group of things. In wheat, it is composed of the two proteins prolamin and glutelin. Other grains also have related proteins. For example, maize and rice have glutens as well, but they are comprised of significantly different proteins from those found in wheat.
Prolamin is one type of gluten. Each plant species that contains prolamin has a different type. Here are some examples of different prolamin proteins found in various plants:
Gluten of one type or another are found in grains. For those with a gluten allergy or celiac disease though, the primary culprits are wheat, barely, rye and oats. There is debate about oats, but I suggest you avoid oats as I believe they are a borderline problem. Many celiacs do poorly on corn as well, and corn is a common food allergy.
Gluten and Celiac Disease
Gluten from wheat, oats, barley and rye is the big culprit for those with celiac disease. You can read more about celiac disease else where on this website. The short version is that if you have the genes for celiac disease, then sooner or later, eating gluten will make you very ill.
Gluten can trigger allergy problems for many people who are not actually celiac. They may experience many of the same symptoms as those with celiac disease (there are some differences). The result is the same: avoid eating gluten.