Inside the casing of the external is a standard 1.8" hard drive.
It has standard SATA Power and SATA Data ports.
The casing has an interface adapter built in. One side plugs into the
standard SATA ports, the other side is the USB port you use.s
If this adapter is not making good contact with the
standard SATA ports you can get these issues. Sometimes you can slightly push and twist the adapter while connected and get it to make good contact temporarily.
You should also try another USB cable to check that it is not something simple like that. Also try other USB ports from your PC.
If you have to give up on the external drive (and it's not still under warranty) you can remove the actual hard drive from the casing and hook it up to the SATA connectors inside a PC just like any other internal hard disk drive.
Doing this may allow you to access all your files directly, or if necessary use data recovery software.
If the drive does not work connected directly it may be damaged or defective.
Let us know.
EDIT: 1.8" drives actually use a Micro-SATA connector, not a standard connector. To hook the drive up to a PC you may need to use a Micro SATA to SATA adapter like this one:
http://www.newegg.com/Product/Product.aspx?Item=N82E16812200788