Though rum has many styles, it can be generally defined as a spirit distilled from sugarcane - whether that is the juice, molasses byproduct from making sugar, cane sugar itself, sugarcane honey, or any mixture of the above. The rules and laws are different per country but the U.S TTB definition is as follows:
“Spirits distilled from the fermented juice of sugar cane, sugar cane syrup, sugar cane molasses, or other sugar cane by-products at less than 95% alcohol …”
Fermentation is a very important and influential process of production in which the preferred sugarcane derivative is combined with water and yeast, causing a chemical reaction which converts sugar into carbon dioxide and alcohol. This sugarcane wine or wash is then distilled in either a single batch or continuous still depending on the style and desired outcome. In rudimentary terms, distillation is the process in which heat is used to collect and concentrate the desirable parts of the fermented mash by condensing their vapors. Fun fact: gluten proteins are too heavy to survive this process and all spirits are naturally gluten free - despite what you may have heard. Additionally, despite common misconception, rum (along with all other spirits) comes off the still sugar free. Remember how all that sugar was converted into alcohol before distillation? There is no universal way to organize the rum category as it is so vast with so much history and variance of style, but our preferred method can be summarized in this graphic.