Sure! I mean, I'm an amateur myself, mainly having become interested in the subject because of my ongoing fascination with wasps. But, being also an old birder, I wanted to develop a general sense of what "kind" of wasp I'm looking at when I see one, the same way I can tell an accipiter from a buteo from an osprey. That's turned out to be a surprisingly complex proposition! So here's what I've learned about taxonomy and cladistics as part of my research into wasps. Again, I'm an amateur, so if any actual molecular taxonomists should happen by, I hope they'll correct the goofs I am surely about to perpetrate.
Taxonomy, a specialization of biology, is the study of how organisms can be usefully classified. A major aspect of that study is the determination of how species are categorized, and how ancestral relationships are recognized. When this is done through molecular phylogeny (ie direct analysis of genetic relatedness and ancestry), the result is a directed graph called a "phylogenetic tree", whose nodes denote species and whose edges denote descent. Such a tree is called "monophyletic", meaning that it contains both, and only, a specific common ancestor and all descendants of that ancestor. Any subgraph from a single node of a monophyletic tree is also monophyletic. In this way, it's possible to establish a relatively unambiguous understanding of descent and relatedness among possibly widely varying organisms.
There are other kinds of classification which often produce groupings that make an intuitive kind of sense in terms of similar characteristics and the like, but which don't accurately represent relatedness. The Linnaean taxonomy is one of these, and the two terms used in cladistics to describe its categories of error are "paraphyly" and "polyphyly".
A paraphyletic grouping is one which contains a single common ancestor, but fails to contain all of its descendants. Many hymenopteran families are like this, such as Crabronidae, which is "all Apoidea that aren't bees" - bees and ants being actually descended from basal wasps, which were themselves descended from an even more basal shared ancestor with modern sawflies. Paraphyletic groupings can be useful, but aren't to be relied upon to accurately describe descent, and are replaced with monophyletic groupings as the necessary research and analysis is performed.
A polyphyletic grouping is one which fails to contain a common ancestor. For example, the European paper wasp (P. dominula) and the eastern yellowjacket (V. maculifrons) closely resemble one another in size, body shape, markings, and much behavior. Based on that, a Linnaean taxonomist might classify them as separate species within a single genus. This (hypothetical) genus would be polyphyletic; while these two wasps are indeed closely related, they're not that closely related, sharing a common ancestor with several other genera including all other social wasps and a sizable fraction of solitary species.
How exactly this all relates to the concept of "species" is hard to really define, at least for me as an amateur, but I think maybe also for professionals in the field. The classical definition, that a species consists of all individuals able to interbreed and produce fertile offspring, and none which cannot, fails to explain cases like wolves and coyotes, lions and tigers, or horses and donkeys - all pairs which are classified as separate species, but nonetheless are able to produce fertile offspring at least some of the time. Phylogenetics offers a distinct species concept [1], defining a species in terms of interrelatedness and innovation in traits as compared with ancestors - I suppose the classical definition could be called "horizontal", while the phylogenetic one could be called "vertical". Notably, the phylogenetic species concept appears to place no significance on interfertility or the lack thereof, which would mean that Equus, Canis, and Felis are not necessarily paraphyletic at all - they might be, but that wouldn't be defined on the basis of interfertility.
In the limiting case, I suppose that means each individual organism could be considered to constitute a unique species, although that is probably nonsense and certainly too granular to be useful. In any case, I hope all this answers your question, and provides some ground for further study. I've certainly learned something today, just looking up references! Thanks for the impetus to further systematize my understanding of the subject.
[1] https://www.oxfordreference.com/view/10.1093/oi/authority.20...
