No it won't have the different drive letter when its back in your old computer.
Take the hard drive out (im hoping you know what a hard drive is), look at the front end of it, youll see some pins (about 8 pins), and if theres a jumper in there already (a little thing that has been inserted around two pins), read the jumper settings ontop of the hard drive. therell hopefully be a diagram displaying where teh jumper needs to go in relation to those 8 pins for the drive to be recognised as a slave.
An easier option is unplugging the wide, usually grey, cable (aka IDE cable) and power cable from the back of your CD drive and plugging them into your hard drive. If you look on one side of the IDE cable youll see a red line, that indicates pin 1. As youre inserting the IDE cable, pin 1 must be closest to where you plug the power into the hard drive. If you only have one CD drive you won't have to worry about jumper settings.