if you want to copy the exact partition to the same size one, use dd:
dd if=/dev/hda1 of=/dev/hdb1
will copy hda1 partition onto the hdb1 partition. if you want to copy particular files:
find -xdev /source | cpio -pmv /destination
which will copy /source to /destination/source.
if you're copying somethign like the /usr partition, then i'd suggest monting the new partition at /mnt/usr and then replacing /destination with just /mnt for example.