"Scientific Naming," aka "Binomial nomenclature," is the official system for giving names to organisms. They usually are derived from Latin. There are really a couple of advantages to the "scientific" naming system, and one disadvantage.
First off, scientific naming is standardized. Each creature has only one scientific name, and each name refers to only one creature. This is of great advantage because it allows precision. While most people might refer to a lot of things as a cat, there are big differences between house-cats and, say, lions. This keeps generalities from skewing meaning.
Secondly, what someone in your neck of the woods might call "stink weed" might be called "skunk vine" in another town, while "skunk vine" might refer to a creeping black and white flower someplace else. Common names are only common to certain areas. Think about how many people call it "pop" and how many call it "soda." Ask for soda in France and you might get soda water instead of Pepsi.
Third, the scientific naming system allows people to see the relationship between animals. Cats make a good example. A housecat and a lion are both part of the family Felidae. This would let you know, just by seeing the name, that the housecat and the lion are relatives. But house-cats are part of the subfamily Felinae, and lions are part of the subgroup Panthera (not to be confused with the cheesy metal band Pantera.) This would let you know that while house-cats and lions are relatives, they are not closely similar; a tiger, though, also in the subfamily Panthera, is.
The big drawback is that the names aren't as cool as "stinkweed" and so people can't remember them. If you said you saw a felis leo in your back yard nobody would know what you were talking about, but if you yelled "lion!" they would.