Loaf sugar is sugar which comes in the form of a solid block, rather than as a granulated substance. Through the early 20th century, the bulk of the sugar on the market was in loaf form, for a variety of reasons. Several sugar producers continue to make sugar this way as a nostalgia item, and in the developing world, it continues to be extremely common.
To make loaf sugar, sugar producers pour hot sugar syrup into a mold which is in the shape of a cone or loaf. When the sugar cools, it can be wrapped and packaged for shipping and eventual sale. The advantage to this shape is that it is easy to handle and ship, because blocks are much less difficult to handle than granulated sugar. Producers also obviously do not need to worry about clumping.
The disadvantage of loaf sugar for cooks is that it can be difficult to handle in the kitchen. It was often hung from the ceiling in the kitchen, although it could also be stored in cupboards. Historically, people used tools known as sugar nips to break chunks of sugar off so that they could use it in cooking and baking. Because of the difficulty involved in obtaining precise measurements, loaf sugar was especially irritating for bakers. It also had to be thoroughly broken up so that it would not clump in baked goods and other foods.
Most loaf sugar on the market today is made with minimally processed sugars, but technically any sort of sugar could come in loaf from, from highly refined white sugar to demerara sugar. A Mexican version known as piloncillo is usually made with lightly refined sugar, and it is readily available in Mexican markets.
The major substantive difference between loaf and granulated sugar, other than the texture, is the moisture content. Loaf sugar is higher in natural moisture, which can in turn play a very important role in baking, where moisture becomes critical. If you have a recipe which calls specifically for this type of, you may want to add a drop of molasses to the recipe if you use granulated sugar, to make up for the lack of moisture. If, on the other hand, you are using loaf sugar in a recipe which calls for granulated sugar, you may need to up the amounts of dry ingredients like flour to cope with the extra moisture.