Serre vs. griffe
The French words serre and griffe both translate to "claw" in English, but they are categorized by the type of animal they belong to. The choice between them depends entirely on whether the animal is a bird of prey or another creature like a mammal or reptile.
Serre
The word serre refers specifically to the talon of a bird of prey, such as an eagle, falcon, hawk, or owl. It is historically related to a French verb meaning to grip or squeeze, which perfectly describes how these predatory birds use their talons to catch, hold, and carry their prey. You will not use this word for mammals or reptiles.
L'aigle a attrapé le gros poisson avec ses serres puissantes.
(The eagle caught the large fish with its powerful talons.)
Le fauconnier porte un gant en cuir épais pour se protéger des serres de l'oiseau.
(The falconer wears a thick leather glove to protect himself from the bird's talons.)
Le hibou a planté ses serres dans la souris avant de s'envoler dans la nuit.
(The owl dug its talons into the mouse before flying away into the night.)
Les serres de ce rapace sont conçues pour percer la chair instantanément.
(The talons of this bird of prey are designed to pierce flesh instantly.)
Il est fascinant d'observer les serres d'un vautour de si près.
(It is fascinating to observe the talons of a vulture from so close.)
Griffe
The word griffe is the standard and most common term for a claw. It is used for mammals like cats, dogs, bears, and wolves, as well as reptiles, amphibians, and mythical monsters. Unlike serre, a griffe is typically associated with scratching, digging, climbing, or swiping.
Mon chat fait toujours ses griffes sur le côté de mon canapé neuf.
(My cat always scratches its claws on the side of my new sofa.)
Le chien de berger s'est cassé une griffe en courant sur le béton.
(The sheepdog broke a claw while running on the concrete.)
Les griffes de l'ours laissaient de profondes marques sur l'écorce de l'arbre.
(The bear's claws left deep marks on the tree bark.)
Le tigre a sorti ses griffes avant de bondir sur sa cible silencieusement.
(The tiger unsheathed its claws before pouncing on its target silently.)
Nous devons couper les griffes de notre lapin ce week-end.
(We need to trim our rabbit's claws this weekend.)
Le dragon de la légende utilisait la grande griffe de sa patte pour se battre.
(The dragon of the legend used the large claw on its foot to fight.)
Summary
In short, you must use serre when referring to the grasping talons of a bird of prey, and griffe for the claws of almost any other animal, especially mammals and reptiles. If it flies and grips its prey, it has a serre, but if it walks and scratches, it has a griffe.