Sugar has been used in Persia and the Arabian peninsula since the 4th century BC; it was introduced to the western world much later, by the Moors when they conquered the Iberian peninsula in the 9th century. It used to be a luxury food, and was sold in rock-solid cream-coloured or light brown blocks; it had to be chipped off and ground in a mortar and pestle. Since then, of course, sugar has become a ubiquitous flavour enhancer, and a very cheap source of simple carbohydrates.
These days we get sugar from a variety of sources, with a corresponding plethora of labels. Sugarcane, beetroot, maple sap, and sorghum are the origin of sucrose; grape or corn sugar gives dextrose; levulose (from fruits) gives fructose; then there's milk sugar, lactose, and malt sugar, maltose. Remember these when you read the labels of processed foods, which may contain a lot more sugar than you think at first glance; they are often all listed separately as if they were many ingredients instead of one.
As for the forms of sugar, granulated or white sugar comes from cane or beet sugar, and is highly refined. Superfine or caster sugar (or castor or berry sugar) is more finely granulated, so it dissolves instantly. You can pulse regular white sugar in a food processor to make superfine sugar. Icing sugar (confectioner's or powdered sugar) is finely crushed white sugar to which cornstarch (corn flour) has been added to prevent clumping. Crystal sugar (sugar crystals, decorating or coarse sugar) has granules about four times larger than granulated sugar, and is used for sprinkling on baked goods; rock candy is even larger. Brown sugar is white sugar combined with molasses to give it a soft texture; it's usually sold as either light or dark, but there's also a very dark or "old-fashioned" style that has a very intense molasses flavour. Southeast Asian cooks use palm sugar, made from palm trees. Raw sugar is the residue that's left after the sugarcane has been processed; in its true raw state it contains contaminants, but purified it is sold as Demerara sugar and Barbados sugar, the latter being the more moist and fine-textured. Turbinado sugar is coarse raw sugar that has been steam-cleaned. Sugar is also sold in syrup form, as cane syrup, corn syrup, golden syrup, honey, maple syrup, molasses, sorghum, and treacle.
Sugar is very high in calories, so these days people often substitute artificial sweeteners like aspartame and saccharine for sugar in their coffee, tea, and even baked goods. The use of these artificial ingredients is controversial, however, especially since the Food and Drug Administration withdrew its approval of cyclamate as a sweetener. They don't seem like a good idea to me, but make your own informed choice. enkidu has a good writeup on the topic here.