The manufacturer is Seagate.

Okay, perhaps the following can work even if the CD drive isn't First Bootable in the BIOS.

On another system go to Seagate's website.

Find their free harddrive diagnostic utility called Seatools and burn it to a CD.

Place CD in system with issues drive and reboot.

Seatools will almost immediately engage and run the tests in this order: S.M.A.R.T., Short DST, and especially the Long DST.

* Get back to us with the results.

SMART Test: Passed
Short DST: Failed
Long DST: Failed
Short Generic: Failed
Long Generic: Failed

I'm going out on a whim and saying this is most likely a bad thing, so I'm going to boot my computer as few times as possible. I'm getting a new external for Christmas, so I want it to last until I will be able to transfer my files... unless there is a way to fix these errors?

Back up everything NOW either to flash drives or burned to a DVD/CDs. Do not wait for that external harddrive. Seriously, your harddrive can crash at any second and data recovery from a crashed harddrive is very expensive.