Some PCB makers will take the CAD files from eagle cad. Send one file.
All cad programs make gerber files. All PCB makers use gerber files. Gerber is the common link.
Making gerber files is a little complicated and errors can happen. You have a file for each layer. (top, bottom, 1,2,3,4) Two file for silks (top, bottom) Two files for solder mask (top, bottom). A file for drills. A file for board size and shape. And more file if you are doing something complicated.