If the USB drive works on another computer it's almost certainly an issue with your USB slot.
If this doesn't work I'll try to help further:
1) Try all USB slots
2) Try another USB device (like your mouse) with all USB slots
3.5" USB hard drives all have AC adapters. 2.5" USB drives use USB power.
If a mouse works on your USB slot, but a 2.5" USB drive does not (and works on another PC) it may be that there is insufficient power on the USB slot. That is pretty rare though.
*It could also be a problem with drivers not being installed or a software glitch, but start where I mentioned and we'll see what happens.