What is a medium write error?

The word 'medium' is used to talk about the hard drive or optical drive. A 'write error' means that when the computer tried to write some data to the disk, it got an error message from the drive saying that the data was not written to the disk successfully.

In either case, there is a serious problem that must be found in order to use the medium. If it is a hard drive, use Disk Utility to check and repair the disk. If it is an optical drive, use a different writable disk.