Everybody is familiar with the dinosaur Tryannosaurus rex, but did you know that it was a tyrannosaurine tyrannosaurid tyrannosauroid? It's true, and you really did read that last line correctly. Understanding how this makes any sense, though, requires a bit more explanation.
Most of us are familiar with the genus, and sometimes species, names of dinosaurs. Tyrannosaurus rex, for instance, is a species of the genus Tyrannosaurus. This means that if another species more similar to Tyrannosaurus than any other dinosaur was found it might be described as a new species of this genus, maybe something like "Tyrannosaurus imperator." The genus and species names are italicized according to scientific convention, but group names above the genus level are not.
Tyrannosaurus did have some close relatives like Daspletosaurus and Tarbosaurus, though, and these dinosaurs all belong together in a group called the Tyrannosaurinae. The Tyrannosaurinae, as a group, were also closely related to the Albertosaurinae, which contains Albertosaurus and Gorgosaurus. Together the Tyrannosaurinae and the Albertosaurinae belonged to a larger group called the Tyrannosauridae.
Yet there are other dinosaurs like Dryptosaurus and Appalachiosaurus that more similar to the tyrannosaurids than any other dinosaurs but that don't fit into that group. They are instead placed in the Tyrannosauroidea, the group containing the tyrannosaurids and their closest relatives.
The names of these groups are not very imaginative and they can easily cause confusion, but it is best thought of as a hierarchy. We know that all these tyrannosaurs shared a common ancestry but some are more closely related than others. By studying what they share in common they can be placed into groups of dinosaurs more closely related to each other than to others which reveals the pattern of tyrannosaur evolution.
Let's take it again from the top. Tyrannosaurus was a genus of dinosaur more closely related to Daspletosaurus than either was to other dinosaurs. Together these genera belonged to a group called the Tyrannosaurinae. The tyrannosaurines likewise shared more features in common with the albertosaurines than with other dinosaurs, and so both groups (Tyrannosaurinae + Albertosaurinae) made up the tyrannosaurids. The tyrannosaurids, in turn, were more closely related to a number of other predatory dinosaurs like Dryptosaurus than other coelurosaurs, so the tyrannosaurids plus this diverse group of tyrant dinosaurs can be placed in the larger group the Tyrannosauroidea.
Each term marks a different degree of specificity. If you call Tyrannosaurus a tyrannosauroid you are saying that it represents the general tyrant dinosaur form shared among a diverse group of predatory coelurosaurs. If you call it a tyrannosaurine, though, you are saying something much more specific about what it looked like and are talking about a much more specific set of dinosaurs.
Admittedly it takes a fairly detailed understanding of dinosaurs and evolution for words like "tyrannosaurine" to make sense. It is not something that can be immediately understood for what it represents. It does fit into a hierarchical categorization of nature, however, and allows this paleontologists to better understand how dinosaurs were related and evolved.