'A corporeal Patronus? ... Your Patronus had a clearly defined form? I mean to say, it was more than vapour or smoke?'
-- Madam Bones of the Ministry to Harry (OP9)
A corporeal Patronus is a fully-shaped spirit animal conjured with the Patronus Charm (OP8, OP16). The Patronus Charm is extremely difficult for most witches and wizards, so a corporeal Patronus that takes the form of a living animal is rare and shows great magical ability (Pm).
An “incorporeal patronus” of smoke or vapour is usually the best a wizard can hope for. Before Harry Potter realized he could use the happy memory of his friends to conjure his silver stag, he could only muster a “semitransparent cloud” of “silvery gas” from his wand (PA12). Lavender Brown and Neville Longbottom were unable to move beyond the vapory stage, and Rubeus Hagrid never learned to do the charm properly (OP27, JKR:Tw) However, someone older and more skillful can us an incorporeal Patronus to purposefully hide their true one, as Remus Lupin blurred his wolf patronus to keep his condition as a werewolf a secret, and because he despised his own Patronus (Pm).
Love can also influence the shape of a corporeal Patronus. Nymphadora Tonks originally had a jackrabbit, which changed over to a wolf Patronus when she fell in love with Lupin (Pm, HBP8). Snape’s Patronus is a silver doe, the same as Lily Evans (Potter), the love of his life (DH19, DH33).
Most corporeal patronuses resemble living animals from the wizard’s country of origin, such as Hermione Granger’s otter or Luna Lovegood’s hare (OP27). Magical creatures such as dragons, Thestrals, or Dumbledore’s phoenix Patronus are rare (GF28,Pm, JKR). Extinct animals are even more scarce, but a wizard named Hedley Fleetwood managed a wooly mammoth (BoS).
Expert Catullus Spangler wrote in the 18th century it was even more rare for a witch or wizard to produce a Patronus that matches their favorite animal (Pm), as with Dolores Umbridge and her cat Patronus (DH13).
Harry received an extra point on his O.W.L. exam for conjuring a corporeal Patronus (OP31). That is ironic since he had earlier been in trouble with the Ministry of Magic for conjuring his Patronus to save himself and cousin Dudley from Dementors (OP1,OP9)
Corporeal Patronuses played a large role in the Battle of Hogwarts (DH31).
Latin corporeus "of the nature of a body," from corpus "body" (living or dead)
Considering the success Harry had teaching Dumbledore's Army how to make their own corporeal Patronuses, it seems odd that the Patronus Charm was never included in the curriculum at Hogwarts. It might have something to do with the Wizarding World legal system, with Dementors being used as the guards of Azkaban prison. If every witch or wizard could conjure a corporeal Patronus, then everyone would have a weapon against the Dementors, and the Ministry might lose control. It would not be the first time the leaders of the Wizarding World were rather short-sighted in their educational plans. ~ SIP