Table sugar has its place in many beer recipes, especially Belgians. It lightens the body to make the beer, in the words of the monks, 'more digestible'. There is also a tradition of using simple sugars in English brewing.
In strictest terms, simple sugars are monosaccharides, which have one sugar unit, but in terms of the basic sugars used as brewing adjuncts I'll also include disaccharides, which are two simple sugars linked together and often ferment nearly as quickly and cleanly. The common ones are below:
Corn Sugar = Dextrose = Glucose = monosaccharide
Maltose = Glucose+Glucose = linked disaccharide
Table sugar = Sucrose = Glucose+Fructose = linked disaccharide
Lactose = Glucose+Galactose = linked disaccharide,
non-fermentable by yeast
Invert sugar = Glucose and Fructose = mixed monosaccharides made by breaking up table sugar with an acid and heat
Some other equivalents:
Brewing grade corn syrup = mostly glucose (kitchen grade might have flavorings and additives you don't want to brew with)
High-fructose corn syrup = invert sugar (corn syrup processed by enzymes to turn half of the glucose into fructose)
Brewing grade rice syrup = mostly glucose and maltose
Candi sugar = invert sugar, partially caramelized for flavor
Honey = invert sugar, though obviously with its own natural flavors
Brown/raw sugar = sucrose, with the natural cane flavoring (brown is white sugar + molasses, raw is cane sugar before the molasses is extracted)
Maple syrup = mostly sucrose
All of these but lactose can be fermented quickly and completely by yeast. Highly purified varieties will lend no real flavor to the end product if used properly, interchangably boosting ABV without affecting body. Less refined sugars will lend their flavors to the end result, so will be more detectable. Corn vs. rice should matter less when you use syrups or sugars than it would if you were mashing the grains with your malt, I would think - you're buying a more refined product and a lot of the unique flavors were removed along the way.