Botanical names allow us to describe a plant with the absolute certainty that it won't be confused with a different plant.
You may have noticed that normal common English names do not provide that clarity. For example, "firecracker" is a confusing name, because it applies to hundreds of different plants. If I call a plant a "firecracker bush" you will not be able to know for certain which plant I am actually talking about.
But if I refer to a plant as Anisacanthus quadrifidus, then you can look that up and find the exact plant I am referencing. Therefore, botanical names are the only means of providing absolute references to specific plants.
The use of these names is called "binomial nomenclature," which is a fancy way of saying "two part names." When you take a plant's genus and species together, you have that plant's botanical name. But I'm getting a little ahead of myself.
Why are botanical names in Latin, instead of English or some other language?
The short answer is: tradition. For long before the current system of naming living things was established, people used Latin to describe plants. Latin has been the language of science for centuries.
Having said that, however, many botanical names are actually in languages other than Latin. Rhododendron, the genus for Azaleas, is from the Greek word ῥοδόδενδρον, which means "Rose tree." Many plants are named after people. The genus Magnolia, for example, was named after French botanist Pierre Magnol. Kniphofia (the red hot poker) was named after the German botanist Johan Kniphof.
Who is in charge of deciding that these names are?
The International Botanical Congress (the IBC) has regular meetings where they make changes to the botanical names in use. Following their meetings they publish updates to the International Code of Botanical Nomenclature, and this code is where we can firmly establish what plants are called.
It's helpful to think of a botanical name as a categorization system. All plants (indeed, all life forms) are organized into the categories: Kingdom, phylum, class, order, family, genus, and species.
All plants are in the Plant Kingdom. From there, they get sub-categorized by phyla, class, orders, and so on. For our purposes, we really only care about the last 2 categories: the genus and species.
Imagine you have a bunch of plants that are similar but not exactly the same. Take mint plants, for example. There is a huge variety of different mints, but they are all mints. So, you can put all mints in a box called Mentha, and that is your genus. Now, within the genus Mentha, you have all the different species of mints, like Peppermint (Mentha piperita), Pennyroyal (Mentha pulegium), or Spearmint (Mentha spicata).
The genus (Mentha) tells you this is a mint. The species (the second part of the name) tells which which mint it is.
What about cultivated varieties that are the same species?
You have surely noticed that there are plants in cultivation that are clearly different kinds, yet they have the same genus and species. These cultivated varieties are distinguished using a "cultivar name." When a plant is a specific cultivar, then the botanical name will include the cultivar inside single quotes, directly after the genus and species.
There is a variety of tomato, for example, called Cherokee Purple. The genus for Tomato is Solanum, and the species name is lycopersicum, so the full botanical name for this variety of tomato is Solanum lycopersicum 'Cherokee Purple'.
I have seen plants that have only one word for their botanical name, like Rosa, Hemerocallis, or Hosta. What's up with that?
When there is a plant that is a hybrid between at least two species, the plant is often represented solely by its genus. This is most often seen in the most popular of plants, like Roses, Daylilies, Hostas and Lilies. In these cases, the botanical name can simply be the genus (in italics) followed by the cultivar name (in single quotes).
Botanical names are easy to figure out, once you know how to read them. The genus is the first word, and what remains is the species. The cultivar, if present, comes after that in single quotes.
Understanding botanical names can help us be better gardeners! When you're talking about a plant, being accurate about which plant it is sure is important. :)