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 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 its life force attached to that species, so if a species were to go extinct the nymphs would die.
Each nymph is able to change into the type of animal he/she is, and only that type. So the bear nymph could not turn into a mouse, but he/she could turn into a grizzly bear, black bear, polar bear, etc.
While in animal form they have all the normal abilities of their animals. So if the nymph is a hawk nymph he/she can fly.
While in regular nymph/person form, they only retain passive characteristics/traits from their animals. So if the nymph is a hawk nymph, perhaps he/she remains claustrophobic in enclosed spaces but cannot fly.