Why aren't common names enough? A blue jay is a blue jay, right? This might indeed be enough, if a scientist was only studying that one species, or staying in the region where this was the only jay. But now suppose you went traveling, and saw different jays? There is the Steller's jay, the scrub jay, the gray jay. But still, they're all jays, right? Yes, but some jays are more closely related than others. Scientific names become important when trying to understand the evolutionary relationships among all these different jays.
Let us begin with the blue jay. Science calls it Cyanocitta cristata. Now, go west of the Rocky Mountains, to where the Steller's jay and scrub jay both live together. The Steller's jay is Cyanocitta stelleri - still in the genus Cyanocitta, so we know it is very closely related to the blue jay. But the scrub jay is Aphelocoma caerulescens - that is a different genus, so we know that it is not so closely related to either of the other two.
This is the purpose of the scientific naming system invented by Linnaeus. Linnaeus was attempting to group plants and animals in a systematic way, with the most closely related kinds groups together. This is why there are so many levels of terms to classify a species. To take the blue jay again, we start with the animal kingdom, since it is clearly an animal rather than a plant, fungus, or microbe. Within the animal kingdom, it is in the class of birds - all those animals with feathers covering the body, and bills instead of jaws with teeth. Within the birds, it is in the order of perching birds. Within this order, it is in the Corvid family, that is, all the crows, ravens, magpies, and jays, because these all have certain features in common and are considered to be more related to each other than to birds in other families. Within the Corvid family, it is in the subfamily of jays, since jays are still more like each other than like crows and magpies. And within the jay family, the genus Cyanocitta is just the blue jay and Steller's jay, since these are more like each other than like any other jays. So, to a scientist, the name Cyanocitta cristata tells more about the bird's relationship to other birds than the common name, blue jay.
The science of naming animals has been called taxonomy, which literally means "arranging names;" but nowadays, many scientists prefer the term systematics, because this better expresses the purpose behind all the names: to show systematically how each kind of animal is related to other kinds.