That is the the definition of species as laid out by Ernst Mayr, and it does have merit. It is a good definition but it gets a bit murkier at times, for example when polar and grizzly bears have fertile offspring, despite the polar being U. maritimus and the brown bear being U. arctos, commonly defined as different species of bear. As such, modern homo sapiens sometimes carry genes from H. neanderthalensis, but they are described as a different species to sapiens.

One issue that arises is that not all offspring between napiens and neanderthalsis were fertile.. The definition comes into question when only half the offspring between 2 ““species”” are fertile, are they different species or not?

There are multiple different concepts for defining species. The long and short of it is that species are not a fact of nature but a useful construct for us to be able to describe it.

Some other factors that are used to distinguish species:

• • The organisms are biologically capable of reproducing, but their genitals are incompatible and prevent it from happening outside of a lab setting.
• • A behavioral difference prevents individuals from different groups from reproducing. For example, several bat species have recently been redefined as multiple species by scientists because they’ve found that they communicate at different frequencies and they aren’t always even capable of hearing the other group’s communication. So even though they can interbreed, they don’t.
• • Geographical separation combined with physical and/or behavioral differences. If two groups never interbreed and you can consistently distinguish them from each other, it might be worth considering them separate species. This comes into play a bit more when it comes to conservation efforts because there is incentive in labeling a unique population as its own species, in order to make it more apparent that the group is endangered even if it is technically capable of breeding with another group that isn’t endangered.
• • Two groups are capable of interbreeding, but have enough differences that hybrids between them are less fit and tend to die before reproducing even if they are fertile. If you have a population of lizards that is good at swimming but bad at climbing and a closely related population that is good at climbing but bad at swimming, their offspring might just be bad at both and struggle to get food. Even if they regularly reproduce with each other, the genetic impact of that on either group is negligible because only the non-hybrids are making babies.
• • Morphological differences. If two things just look really different, we can call them different species. This is especially true for organisms that don’t reproduce sexually and for fossil species. There’s no way to test if a Tyrannosaurus could have a baby with a Velociraptor, but odds are it couldn’t. Truthfully it gets even more complicated with fossil organisms and can sometimes come down to scientists saying “the ankle bone developed a little knob around 76 million years ago and 10 million years is a long time to be considering these fossils all of the same species, so we are going to use that ankle bone knob as a marker of a new species distinct from the species without the knob that came before it”.