‘Rum’ is a catch-all term for every style of the spirit. However, when people talk specifically about 'rum' in contrast to 'ron' or 'rhum', they are usually referring to molasses-based English-style rums produced in former British colonies. Distillation usually occurs in a pot still and the finished product tends to be relatively heavy in body, rich in flavour and strong in aroma.
If you see a bottle labelled 'Ron', it is likely to be a molasses-based 'Spanish-style' rum produced in a former Spanish colony such as Venezuela, Puerto Rico or the Dominican Republic. Distillation is more likely to take place in a column still. Compared to English-style rums, ron tends to be lighter in body and less strong in flavour and aroma.
Rhum is derived not from molasses but from fermented, fresh-pressed sugar cane juice grown specifically for this purpose. The term is short for 'rhum agricole' (meaning 'agricultural rum') and this style of the spirit was originally distilled on French Caribbean islands such as Martinique. Rhum has a fresh, earthy and grassy flavour with a less syrupy mouth-feel compared to some molasses-based rums.