They have a telepathic/empathetic connection with nature and other nymphs. They also have a special, stronger, connection with other animal nymphs
As they are nymphs they do not age, remaining eternally young.
Their senses of smell, sight and hearing are more enhanced, enabling them to sniff out monsters, track down demigods, etc.
They are able to summon animals to aid them, they have the best control over their own animal type
Specific to each Animal Type:
Each type of animal nymph has their life force attached to that species, so if a species were to go extinct the nymph would die
Each nymph is able to change into the type of animal they are, and only that type. So the bear nymph could not turn into a mouse, but it could turn into a grizzly, black bear, polar bear, etc.
While In animal form they have all the normal abilities of the animal. So if they are a hawk they can fly.
While in regular nymph/person form, they only retain passive characteristics/traits from the animal. So if they are a hawk, perhaps they are claustrophobic in enclosed spaces, but they cannot fly
Varies between species