Use the xcopy command.
Specify your source as a folder (backslash on the end) and the command will operate recursively with the switch /E
xcopy "c:\source\" "c:\destination\" /E
The folder contents will be recursively copied to "c:\destination\"
Finally, use the ren
command to rename your folder:
ren "c:\source\destination" "copiedfolder"
Your finished batch file will look like:
xcopy "c:\source\" "c:\destination\" /E
ren "c:\source\destination" "copiedfolder"
As stated in another answer, the destination folder has to exist. If it does not, consider the mkdir command!
As a note, try entering xcopy /? for extended help on the command. /? works with many standard windows commands.