Do i need to purchase an external hard drive in order to transfer my photo library to another computer?? Is there any other technology that is cheaper?
You could physically move the HD on your current computer and install it in the new computer.
Do you have backups of your photos and catalog file on a different disk? If so, you could physically take the backup HD and install it in the new computer. If not, then investing in an external HD for backups (and transfers) is highly recommended. You are living extremely dangerously if you don't have backups. I wouldn't even bother doing the transfer until you are sure you have backups on a different disk.
I suppose you could use WiFi, Ethernet or a cloud service like DropBox.
You could borrow an external HD from a relative or friend.