People of Asian or European ancestry have 1-2% of their DNA from Neanderthals. People of African ancestry usually have no Neanderthal DNA.
But consider the identical ancestor point [1] (IAP) for humans. The IAP is the most recent point at which each individual then alive either (1) has zero now living descendants or (2) every now living person is one of their descendants.
Most estimates put the human IAP at less than 20k years ago (some estimates much less).
Neanderthals went extinct around 35k-50k years ago.
I have Neanderthal DNA (23andMe tells me I have 240 markers with at least one Neanderthal variant, about average for 23andMe customers). Therefore I have a Neanderthal ancestor who must have lived 35k or more years ago.
Unless the estimates of the human IAP are so far off that it was actually before my Neanderthal ancestor lived, I therefore must have an ancestor who was alive at the human IAP and who had Neanderthal DNA.
Since that ancestor of mine has at least one living descendant (me) and was alive at the IAP, everyone now alive must be a descendant of theirs. Thus I can conclude that everyone alive now has Neanderthal ancestry--including all those Africans with no Neanderthal DNA.
This at first seems contradictory, so what's the deal?
Is it simply that humans have about 30k genes, your children each only get half of your genes, and so after about 16 generations it is possible for someone to be a descendant of yours but not have any genes that came from you?