A binomial name is a name composed of two parts using Latin. Binomial nomenclature is the name of the system which names living things in this way; the system was developed by the scientist Linnaeus, among others, to help classify animal and plant species into groups and make them easier to identify and study. (Quite often if people want to know the binomial name of a plant or animal they just ask 'What is its Latin name?' which basically means the same thing).
The first part of a binomial name tells you what genus the species belongs to. The genus is, roughly, the branch of a family to which the species belongs. For example, a boa constrictor belongs to the boa branch of the snake family; constrictor is the name of the distinct species to which it belongs. Likewise humans belong to the genus homo and the species sapiens.
You can see some examples of binomial nomenclature in this table, which also shows you how binomial names are written - only the first letter of the first word should have a capital letter.