What do you call the car who pick the trash up?

Trash Truck

If you want specifics, it'll depend on the method the waste is picked up. In a residential neighborhood, you'll most likely see a compactor. They can be front loading (the ones that pick up dumpsters and tip them up to dump into the bin), side loading (these pick up trash cans with an arm, dump them into the can from the side, and place them back down), or rear loading (a person grabs the trash can, loads the contents into the back, and replaces the can).

If it's picking up a roll-off container, that's a roll-off truck.

For scrap wood and other construction materials in bins, that could be a couple different trucks. Some trucks have a boom and grApple which will pick up the contents and drop it into the can. Some services use simple dump trucks, which can be either hand-loaded, or loaded by machine (such as a backhoe or skid steer loader).