In short, you're likely to be in way over your head. Buuuuut I think we were all there at some point, and struggled our way out of it. Somethings I don't recommend you do... do not take the drive itself apart. Do not freeze it. Do not microwave it. Do not tap with screwdriver or hammer. Do not shake (it) violently. I had a friend find all these suggestions (and even try a few) when his external drive died with all his files on it.
Most of these external drives are just a plain 2.5" or 3.5" SATA drive with a SATA to USB adapter stuck on it. If the controller USB adaptor board is bad, you might be able to swap it off a same-model product for a known good one. The reason I say same model with such emphasis is that sometimes there is proprietary information on that board needed to access the drive.
If it is the drive itself, try pmagic or sysresccd. They are live boot linux distros that come packed with data recovery tools. Create a block-based image of the original drive on a second device (you might hear this refered to as "dd" the device), and work to recover your data off the image, not the original.
If I recall right, testdisk has the ability to search for and recover lost partitions. If it is just the partition tables that are bad, your data is still there in normal form, but your computer doesnt know there is a partition there with that data in it. If you can recover the partition, you may be in luck and able to access the file system in the usual fashion.
If not, you may need to run an actual data recovery tool, checking the disk for bad sectors, recovering them where possible, and recovering files potentially from the raw data they are composed of. It really depends how bad you need that information back. Small files will be easier (more likely) to recover successfully than larger ones - the bigger the file, the better the odds that something of it is missing.
Edited for clarity with regards to controller vs usb adaptor. Did not want him getting confused for the controller board on the drive itself