Normally, files are verifies after they are written by re-reading the file from disk and comparing the result to the original file.
This is important when copying to a floppy disk, because it is possible that the file may not be written correctly.
When copying to a hard disk, this step is not necessary, because hard disks are more reliable.
In situations where you are copying between hard disks, or between a hard disk and an AppleShare server, this option may be turned off, to speed up copying.
Verification of the written file is not done when no check mark appears next to "Verify files" in the Options menu.