Plants have common names and botanical names. The common names are sometimes really interesting and descriptive, but botanical names allow for accurate identification of a plant. The same common name can sometimes be used for different plants, and the same plant often has different common names in different places.
Botanical names must have at least two parts for accurate identification. This is known as the binomial system of plant naming, or nomenclature. Previously, long descriptions were used to identify individual plants. The binomial system condensed this to two parts, the genus and species.
The genus or generic name is the first part. A genus is a group of related plants belonging to the same family. The generic name is written with a capital to start with, and is in italics. The second part of the name is the species or specific epithet. A species is a group of related plants in a genus, and these can generally breed with each other. The species name doesn't have a capital, but is also in italics. Specific epithets are often really descriptive and can give you interesting information about the plants.
Latin is used for the botanical names as this was the language that was widely recognised and understood at the time. Very often the names are just 'Latinised', and Greek words are also incorporated.