For my science homework, a question came up asking how many species can have the same scientific name. I'm pretty sure that each species has a different scientific name, but just to be sure...
The scientific name (better known as latin name or binomial name) of a species is unique to this species. No two species can have the same latin name. Also, a single species cannot have two different latin names. But of course, mistakes happen and we don't seem to bother too much about them (esp. when the two species are very unrelated; see What instances are there in which two species share the same binomial name? (thanks @fileunderwater and @HRA).
Of course, the above is true as long as there is agreement in the limits of the definition of a species. In absence of such general agreement, there can be some issues. Consider for example, Homo neanderthalis who is often called Homo sapiens neanderthalis. See the post How could humans have interbred with Neanderthals if we're a different species? for more information about the definition and delimitations of species.
Have a look at wikipedia > Binomial nomenclature for more information.
This is not to try and replace Remi.b's great answer, but just to supplement it for any later visitors: biological nomenclature is controlled by international codes which regulate how and what species can be named.
For example, the ICN (International Code of Nomenclature for algae, fungi, and plants) governs the naming of species and groups within those taxonomic groups, while the ICZN (International Code of Zoological Nomenclature) governs naming of animal species.
One piece of trivia about the two: tautonyms, or tautonymous names, are species names where the genus and species epithet are the same, for example Rattus rattus (black rat) or Vulpes vulpes (red fox). These are allowed under the ICZN, but are explicitly prohibited in the current edition of the ICN (the Melbourne Code published in 2011).