What model of Lacie external drive is it?
You normally can't physically connect the power connector backwards, unless you use considerable force to force it in / on .
Have you dropped the external drive enclosure ?
You MUST provide sufficient power to external hard drive enclosures, or they won't work.
If it's a model with a 3.5 drive, you MUST use an external power adapter plugged into the external enclosure, and it must be working.
The USB port you plug it into MUST be able to supply the full USB max spec 500ma.
If the USB cable you're using did NOT come with the drive, it may NOT be sufficient.
If it's a model with a 2.5" drive, you MAY need to plug it into TWO USB ports, or plug into one USB port and use a 5v power adapter plugged into the enclosure if it has a jack for that. .
Troubleshooting USB device problems including for flash drives, external drives, memory cards.
See Response 1:
Check that out first.
Rarely, not all the ports on the back of a desktop case may be able to supply 500ma each.
If you have a desktop computer, Note that I answered a Topic on this site recently where a guy had an external drive, which does require the full 500ma, connected to a port on the back of a desktop case - it would not work properly when a webcam was in the port next to it, but it worked fine when the webcam was unplugged. Ports on the back of a desktop case often have two ports connected to the same USB controller module that are ports one above the other - you could try connecting the cable to one of those and leaving the other un-used.
If the external enclosure is damaged, it's likely the drive inside of it is still okay, more so if it's a 3.5" drive and if the external drive has not been dropped.
You can buy new external drive enclosures starting at about $20 for a 2.5" hard drive, or starting at about $25 for 3.5" hard drives. It must be for the same type of hard drive - EIDE (PATA), or SATA, or one that can be used for either.
I recommend Vantec ones.