When not using his image inducer, Nightwing is a figure in black armor with a blue V across the chest. While it covers all of his skin, you can see that his proportions are off and he has a tail. His hands and feet have three long agile digits. A tactical harness is designed for quick access to a number of devices. Wrapped over his shoulder is a weighted chain that is his signature weapon.
Only the rare people that he trusts get to see Kurt Grayson outside of the armor, and then only is a secure location. Without the suit, his odd proportions are more obvious. He has dark blue fur all over his body, yellow eyes, and small fangs. He's usually smiling and joking around when he's in such a safe space. Still, signs of danger cause him to react quickly. A common trait among people connected to the Bat.
History (subject to change):
Kurt doesn't know about his parents. As a newborn he was found abandoned at the circus. Being blue and having a tail explained the abandonment, but circus folk tended to make a place for outsiders. John and Mary Grayson took in the baby and named him Kurt. He was about the same age as their son Dick. It wasn't ever a formal adoption, as he was never registered.
Kurt grew up very talented and joined the act once he was old enough. He wore a full suit, skin-colored mask, and hid his tail in the act. He still looked a little odd, but up in the air it was hard to tell why. Kurt accepted this as that was all he knew. But hard times were ahead as the circus went into debt. Eventually the mobsters took their revenge with a massacre of the circus folk and customers. Kurt learned to teleport when the bomb went off. He tried to hide, but as Batman was looking for evidence, he was found. Until he teleported away. It took a while for Batman to calm Kurt down and explain what happened.
Bruce knew what it was like to be an orphan of violence. And he knew that a mutant wouldn't do well in an orphanage, so he invited Kurt home. Bruce kept this secret because Kurt didn't want attention. But Kurt wanted to learn to fight crime. So later, Kurt became Raven, sidekick to Batman. He grew up as a vigilante. Batman trained him to use his powers and how to deal without any powers.
But eventually people need space. Kurt struck out on his own to work independently.