Get a simple answer but without it there is no possibility of a [b]Simple Answer.[/b]
You need to tell us what the OS is on the computer you are currently using and what was on the Old Computer. If it is Windows you may have an issue if you have Vista with a Old FAT8 or 32 Partition as Vista can not read these and is unlikely to read a FAT32 Partition as well. Also if there is a NTFS Partition involved and you are attempting to read it with Vista the new security built into Vista may prevent you reading the Drive as well..
If the Drive came from a IBM Think Pad the Security Chip on the M'Board no longer being present will also prevent the ability to read the HDD unless you are using the same model Think Pad.
But you first need to look at the actual HDD it needs to be set to Master or Cable Select if it is a 3.5 inch IDE Drive and you need to remove the jumper all together if it is a 2.5 Inch IDE Drive.
Supposing that you are using Some Form of Windows have you had a look in the Drive Manager for the USB Drive? Open the Control Panel in Classic View then left click on Administrative Tools then left click again on Computer Management and when that window opens left click on management and post back with what is shown about the USB Drive.
Of course you need to have the USB Drive inserted before doing any of this.
Col