Who gives an organism its scientific name?

The first person to discover and describe the species properly gets to name it. The names are in Latin because that language never changes. The genus name is always a noun telling what the organism is, like Canis for dogs and Felis for cats. The species name is an adjective describing the organism, like Bos bison, the cow that is a bison, for bison. Here is a good Wiki-article on naming organisms, too.