I realize this question is very old, but there is an important point missing from the current answers: That is the difference between the nomenclature and the taxonomy.
By "taxonomy" I mean the understanding of how organisms are related to each other. Key questions here would be whether taxa sharing some group name (the same genus, for example) form a clade (i.e., there exists an ancestor whose descendants include all and only members of that genus), and whether particular taxa or specimens are the same species or not.
By "nomenclature" I mean the names applied to each taxon and groups of taxa.
The key point is that nomenclature exists within some accepted taxonomy, and that when name changes occur, it's usually because of a conflicting or updated understanding of taxonomy. The rules of nomenclature are comprehensive and quite rigid, but for taxonomy there are no rules at all. Each researcher is free to adopt a taxonomy based on their own understanding of the evidence, however within the taxonomic framework they choose, they must follow the rules of nomenclature.
A "synonym" refers to a named species that the current author considers to be a part of another, earlier named species. For example, the white-veined wintergreen, Pyrola picta Sm., was described in 1819. Later, the cryptic wintergreen, Pyrola crypta Jolles, was described in 2014 when the authors determined that molecular evidence and subtle morphological features marked some P. picta as distinct.
Botanists can (and do) have good-faith disagreements about whether these represent two species. There is no over-arching authority that makes that determination. Botanists who think they are different will use both P. picta and P. crypta, respectively. Those who think they are best considered a single species must use the name P. picta for all of them, and list P. crypta as a synonym.
The example previously given of Drosophila melanogaster possibly being renamed stems from a different issue. Each genus name is pinned to one particular species (the "type species"). In the case of Drosophila it is D. funebris. This means that if the genus Drosophila is recognized as a genus, then it must contain the species D. funebris, but not necessarily any other species. What probably happened is that a new family tree (phylogeny) of Drosophila and related genera was constructed, and D. melanogaster came out as being more closely related to flies in a different genus, than to D. funebris.
Ideally, all the members of a genus would form a clade, because, for example, it can be confusing to have Drosophila melanogaster be more closely related to some other genus, than to other members of Drosophila. So to remedy this, some fly taxonomists suggested reshuffling these taxa to make more neat clades and less of a jumble (D. funebris though, would have stayed in Drosophila).
Normally this happens and nobody notices except for the few researchers who work on that group. Here, however, scientists from all over the world in a dizzying array of disciplines have studied and published results related to "Drosophila melanogaster". It would have lead to huge headaches and confusion to rename this iconic species. In this case, a formal proposal was made to create an exception to the normal rules by changing the type species of Drosophila from D. funebris to D. melanogaster. That way, as long as Drosophila is recognized, then D. melanogaster would also exist.
One final point is that @nico's statement "surely there is no correct name," is incorrect. In the world of nomenclature "correct" is a term of art. For a particular taxonomy there is exactly one correct name. It's just that the taxonomy is subject to the interpretation of the researcher or author.
