Biologists organize all living things by placing them in a tree based upon their relatedness. For macroscopic organisms the main limbs of the tree are the Kingdoms -- Animalia (animals), Plantae (plants), and Fungi. The largest divisions under the kingdoms are phyla, which in animals correspond to distinct body plans. Thus arthropods, with an exoskeleton made of cuticle and jointed legs and antennae, are one phylum, and chordates, with a notochord that runs along the back and which have internal skeletons, are another phylum. Most of the animal phyla are various types of worms. Each phylum is divided into classes, which are divided into orders, which are divided into families, which are divided into genera, which are divided into species. Organisms of the same species are considered to be of the same "kind" -- they can interbreed and yield fertile offspring, and in nature rarely or never interbreed with other species. Of course in paleontology the determination of species is somewhat subjective and based on morphology, we can't know what was breeding with what.
Each species is given a binomial name that consists of its genus (capitalized) and its species (in all lower case). The same species name can be used with many different and unrelated organisms, but the combination of genus and species is required to be unique. Thus Cedaria minor is a species of trilobite but Vinca minor is a species of flowering plant.
Sometimes you will see a "binomial name" with two lowercase parts, e.g. Cybeloides loveni girvanensis. The first lowercase word is the species and the second is the subspecies. Thus here the genus is Cybeloides, the species is "loveni" and the subspecies is "girvanensis". Sometimes you will see a "binomial name" with two uppercase parts, where the second is in parenthesis, e.g., Scabriscutellum (Rheiscutellum) lahceni. In that case the parenthesized name is the subgenus. So here we have genus Scabriscutellum, subgenus Rheiscutellum, and species lahceni. I show the subgenus and subspecies in the binomial names but do not have separate levels for them in the taxonomic tree.
Ever since the development of evolutionary theory it has been understood that the taxonomic tree should reflect the evolutionary line of descent of the various species. This leads to the idea of a clade -- a group of organisms that consists of a common ancestor and all its lineal descendants, and represents a single "branch" on the "tree of life". Ideally each phylum, class, order, family, etc. should represent a clade. I say ideally, because as our understanding of evolutionary history changes we find that some historical families, orders, etc. turn out not to be true clades.
To reflect the long history of evolution biologists have found it necessary to interpolate extra levels in the taxonomic tree -- thus we have superfamilies that are above families but below suborders, which are below orders. Even so, many biologists find any division into a fixed number of levels to be too artificial and constraining, and just use as many levels of clades as they see fit.
I stick to the classic levels of kingdom, phylum, class, order, family, genus, and species, with the sub- and super- levels in-between. This does cause problems with the phylum we humans are in -- the history of Chordata goes from chordate worms to jawless fish to early jawed fish to Sarcopterygii (lobe finned) fish to Tetrapodomorpha to early tetrapods to Reptilomorpha to Amniotes to Synapsids to early mammals... The Sarcopterygii are already down to the class level, yet traditionally mammals or at least Synapsids generally are at the class level. So there is a break in the tree if you stick to the traditional fixed number of levels.
In paleontology there are often things we don't know how to classify. For example, there is little consensus on how the strange organisms that lived in the Ediacaran period, just before the Paleozoic Era began, relate to living organisms. Many of them may belong to phyla that don't exist anymore. I just place them in the artificial "phylum" called "Ediacaran" under Animalia. This is a placeholder that doesn't even pretend to be a clade.