Admit it: As a fan of the Harry Potter series, you’ve probably taken at least one personality quiz to find out what your Patronus is. For those of you not familiar, a Patronus is a type of advanced magic—according to author JK Rowling, it's "One of the most powerful defensive charms known to wizardkind ... it is the only spell effective against Dementors."
In Harry Potter and the Prisoner of Azkaban, Professor Lupin tells Harry that a Patronus is “a kind of Anti-Dementor—a guardian which acts as a shield between you and the Dementor,” as the terrifying creatures were what Harry needed protection from.
When casting the Expecto Patronum spell, a gifted wizard will conjure up a corporeal Patronus in the form of an animal, which protects them. (Incorporeal Patronuses are a thing, too; according to Rowling, they "[resemble] a mass or wisp of silvery vapour or smoke." However, this type of Patronus "is not a true Patronus and while it will give limited protection, it cannot provide the defensive power of the corporeal Patronus.")
The type of animal varies from wizard to wizard; Harry’s Patronus, for example, is a stag, which we see for the first time at the Great Lake toward the end of Harry Potter and the Prisoner of Azkaban when dementors come for him.
Rowling has revealed the Patronuses of many of the characters in the series, and even declared her own on social media. While it's a lot of fun to find out which Patronus each character has, knowing that Hagrid doesn't have one—a fact Rowling revealed during a 2015 Q&A—is actually quite upsetting for fans.
According to Rowling, "The Patronus is the most famous (and famously difficult) defensive charm"—so much so that "The majority of witches and wizards are unable to produce Patronuses and to do so is generally considered a mark of superior magical ability." Given that Hagrid was expelled from Hogwarts, it's likely that he simply isn't a gifted enough wizard to cast the spell. Which doesn't mean that he can't at least take one of those personality quizzes.