What's up with the crow?

Depends if you're talking about in the livres ou the TV series.

In the TV series, Damon controls the corbeau, corneille as a pet. It was killed pretty earlier on in the series when Zach was also killed.

In the books, however, Damon IS the crow. vampires who are very powerful have the ability to morph into a specific animal form.

the corbeau, corneille is damon the plus he takes human blood the plus powerful he gets!! and has different powers!

In the TV montrer Damon ate it.

the corbeau, corneille is when toi know when damon is going to montrer up and the the foge when your on the grond

In the books, Damon takes on the form of a crow. It's part of his Power from drinking human blood.

The plus power a vampire has, the plus appearances he/she can take on. plus than two is highly unusual. The only known vamp that could do that was Katherine. She could turn into a white kitten, a white owl, and a white tiger. The corbeau, corneille comes into play with Damon. He is powerful enough that he can take on the form of a crow. He can also turn into a black wolf. Stefan can only turn into a falcon. But yeah, so Damon is the corbeau, corneille in the livres MDR :)
BUT! in the show, I think he just controls the corbeau, corneille instead of BEING it. Does that make sense?

Remember when we first meet Damon in season one in Stefan's room? The corbeau, corneille flies into his room, and Stefan turns around (I didnt like Stefans Hair there... too short), Damon is waiting outside of the door, and the corbeau, corneille would not have had time to fly back to transform into Damon. :)