A corset has been many things over the years, and the subject is one for which many books have been written. For the purpose of this discussion, I will stick to the modern corset as it has existed for about the last 100 years or so, since the incorporation of the two piece steel busk.
A corset is a close fitting garment designed to minimize (usually) the waist while enhancing the bust and hips. The modern corset closes in the front with a busk (see Inside an Enchanted Corset), and laces in the back, though corsets do exist with side lacing, zippers, buckles, and other variations. The key part here is that the corset is made smaller after it is donned, reducing the size of a body part (or parts).
Boning, preferably of steel, is used throughout the corset to keep the garment from rolling, buckling or bulging under pressure. A steel busk or steel boning in the front flattens the stomach, and again in back to prevent pinching.
Fabric selection falls under the category of "the sky's the limit." There are a few limitations, however, if one wants a corset that is restrictive by nature: the fabric should not stretch (i.e. needs to be woven), and if the corset is to be worn on a regular basis, the construction materials should be densely woven and as sturdy as possible. Coutil is the preferred material of choice for the lining and, in the case of waist training and daily wear corsets, the outer layer as well.