Have you tried the drive in different USB ports on your computer? The computer should make some sort of noise when it detects a USB device (or Esata) being plugged in. When you power on the external drive you should also hear the drive spinning up. Does the drive make any odd clicking noise?
Try plugging it into different ports, as mike said. If that doesn't work, plug it into a different computer. If it can be seen by another comp then it's something to do with your comp. Also, if you actually do notice a clicking noise, that means your hdd is failing/failed. In which case try your hardest to recover the data on it.
Clicking noise is bad, it's a hardware issue with the drive. At times a board swap can do the trick, need to check ebay or somewhere for an exact match. May not fix it though, if it does not, you can try a drop of about 3 inches flat on a desk. If that does not work, putting the drive in a freezer for several hours wrapped in a paper towel, then try it. Those at best will get you some time to get the data out, although how much time is unknown. Those things work very rarely.