Plain sugar is a granulated sweetener that comes in fine crystals, about the size and shape of salt crystals. There are different degrees of fineness for sugars and they all have different uses.
Superfine sugar is regular (granulated) sugar that has been ground more finely. It is sometimes sold as baker's sugar and is also known as castor sugar. It can be used as a substitute for regular sugar and may produce a slightly finer crumb(texture) in the finished product than regular sugar would, a result that is preferred by many bakers. Regular sugar, however, should not be substituted in the same way. Superfine sugar is almost always called for in recipes that are very light, like meringues and angel food cakes. It contributes to a light, delicate final product because the small grains dissolve into the mixture very quickly, while a coarser crystal might not be incorporated as fully or as quickly. You can make superfine sugar yourself by putting coarser sugar into the food processor and whizzing it around several times to break up larger crystals.
Powdered sugar, or confectioners' sugar, has been completely pulverized into a very fine powder and often has a small amount of cornstarch added to it to keep it from clumping together. It is much finer even than superfine sugar, so it dissolves almost instantly into a mixture. This makes it perfect for making glazes and frostings smooth and sweet. It is also used in some recipes, particularly those where a melt-in-your-mouth texture is desired in the final product, like shortbread. It cannot be substituted with either regular sugar or superfine sugar.