Until recently the argument was over whether they should be classified as a separate species to modern man (Homo sapiens) as H. neanderthalensis or were they merely a Euro-Asian subspecies (H. sapiens neanderthalensis) with modern humans appropriately classified as H. sapiens sapiens. The idea that they may have been directly ancestral to H. sapiens has all but disappeared, except in Creationist parodies of human evolution.
Then just as the picture seems to be clarifying, it can become even more confused by the discovery of something previously unknown and unexpected. True to form, a previously unknown group of hominids living at least 41,000 years ago in eastern Asia was recently discovered. Named the Denisovans, they appear to be close to but distinct from Neanderthals and yielded enough DNA to make an even more intriguing discovery - many ethnic groups in South-east Asia have some Denisovan genes, showing beyond reasonable doubt that Denisovans, while being genetically distinct from H. sapiens, interbred with H. sapiens in South-East Asia.
Now we find that Neanderthals also probably interbred with modern humans outside Africa, so that modern non-Africans have somewhere between one and four percent Neanderthal genes whilst African peoples have almost none. The interbreeding may not have been frequent. Some have suggested that to contribute one to four percent of our DNA, Neanderthal/Modern human interbreeding need only have occurred about once every thirty years, though there are tantalising suggestions that mixed communities may have existed in Portugal as recently as 24,500 years ago.
A picture of geologically recent human evolution is thus beginning to emerge starting in Africa where we know that the major evolutionary work was done, resulting in an almost modern hominid and undoubted member of the Homo genus, H. heidelbergensis which expanded its range out of Africa and up into Europe and Asia some time around 350,000 - 600,000 years ago where is diverged to become H. neanderthalensis in the west and the (as yet scientifically unnamed) Denisovans in the east. Meanwhile the African branch of H. heidelbergensis evolved into early H. sapiens by about 200,000 years ago.
Then, about 50,000 - 60,000 years ago, a second migration out of Africa occurred, consisting of maybe only a few individual H. sapiens from which all non-African humans are descended. As their population increased and their range extended they came into contact with Neanderthals in Europe and Denisovans in Asia and interbred with both, giving rise to regional variants of the basic African theme.
So, depending on where taxonomists draw the (always arbitrary) line designating species, sub-species and geographical variants, we now have a picture of either three species interbreeding at different points in their overlapping range, or a single species existing with a considerable range of variation within its DNA between different geographically isolated populations. The DNA differences suggest the former.
In fact, what is beginning to emerge looks remarkably like a 'ring species' existed for a considerable period in human evolutionary history, resulting, eventually in a single species with one - the Neanderthals - eventually becoming extinct and the other - the Denisovans - possibly merging with archaic H. sapiens to be the ancestors of several modern South-east Asian and Australasian peoples.
But of course this is to be expected as a species extends its range, as humans did when they emerged from Africa in successive waves over a period of some 600,000 years. A 'ring species' is a case of incomplete evolution or an example of evolution in progress, and that is exactly what H. sapiens was at that time in our evolutionary history.
Hence, H. heidelbergensis, H. sapiens neanderthalensis, H. sapiens and the as yet unclassified Denisovans are the 'transitional fossils' that Creationists dread being discovered.
Personally, I love the thought that Europeans, for all their arrogance and cultural racist supremacism, may be the result of a cross between a human being and a Neanderthal!
See also update to When The Conclusion Is Sacred Facts Must Be Ignored.