Alumina is a metallic element related to aluminum oxide but in the form of a powder. It can absorb liquids very easily and is commonly used as a desiccant. In a filter, its absorption capabilities work better at holding in unwanted chemicals like fluoride and chlorine than water, so it excels at capturing these compounds while letting most water pass by.
RO stands for reverse osmosis, a filtration system that works to separate all particulars from water, including larger chemical compounds like chlorinates and fluoride. It uses a thin membrane that only allows water to seep through while blocking other particles. The downside to this is that these particles accumulate on the other side of the membrane, which must be replaced occasionally to help it keep working.
Distillation is a simple process where water is boiled before it is used. Filter systems that use distillation are highly effective because, as the water boils and evaporates, heavier pollutants stay were they were. Distillation systems catch the steam and divert it back to a liquid state where it can be used without carrying the chemical pollutants.
Carbon filters use activated carbon to capture chemicals, much the same way activated alumina does. However, while carbon filters can easily remove odors and chlorine, they are not efficient at removing fluoride. If you want to remove fluoride as well, you will need to install a secondary system or switch to a different version. Carbon filters are attractive for their cost effectiveness and purification properties.