This is a scenario where you have an external HDD (or “pen” drive) connected via USB to the PI. The SD card will be just used for the boot sequence then everything will work from the external HDD, I find the system to work faster, smoother and more stable this way (at least with my v1 PI).
First thing is to normally install the Raspbian system in the SD card. There will be 2 partitions on it. A first small fat16 partition labeled ‘boot’ with the lba flag set, and a second ext4 partition. So, on your external HDD you will need also a ext4 partition. I advice you to to make 2 partitions in the HDD, one for the OS, programs and files, and the second (like a /home/ or /data/) for personal files, backups, etc. In the event of a major catastrophic malfunction, this configuration offers a bigger safeguard to your personal data.
If you need assistance to partition and format your HDD probably its a good idea to stop here.
Mount and sync the second SD card ext4 partition with your external HDD ext4 partition
sudo rsync -avHx /mnt/sdcard/ /mnt/hdd
Next step is to mount the SD card boot partition. And change in /cmdline.txt (adjust sda1 to your HDD partition as needed).
You can actually delete de SD card system directory (or not) it works the same. Also don’t forget to add an entry in /etc/fstab for mounting the second partition in the HDD in case you have it.
Good luck with your PI.