External hard drive repair and diagnostic steps
1) For the first test, you'll need to be in a relatively quiet area. If your environment is noisy, a bathroom with an outlet in it might suffice. Making sure that the power switch is on, plug the power supply into the wall and then into the external drive. Listen to the drive with it near your ear. Do you hear the whining of the drive spinning up? If you aren't sure, have somebody else listen.
A - Does the drive spin up? Great! Proceed to step 3.
B - Does it not spin up? Double check that the power switch (if there is one) is in the proper position (toggle it if you're not sure) and that the power outlet works. Still not spinning up? Proceed to step 2.
2) The most likely reason that your drive does not spin up is that the enclosure for the hard drive has failed, or the power supply for it has failed. This is good news! A new enclosure should fix it. That will be covered in step 4.
3) Listen more carefully for the next two steps
A - Does the hard drive make a few short clicks after it starts up, and then quiet down? That is good. That means the hard drive is starting up properly. Turn off (or unplug power from) the drive. Now, plug in the USB cable (which is not plugged into the computer, remember) and power it back up. Does it still spin up like before? If not, try a new USB cable. If it does, then try plugging it into a different computer. Perhaps there is a problem with your computer causing it to not communicate with the drive. If it still doesn't work, proceed to step 4.
B - Does the hard drive make loud, harsh double clicks every few seconds? If so, I have bad news. The drive has failed, and most likely taken all the data with it. If it has the 'click of death', consult with a local data recovery company. It might be salvagable if you have the funds. It can get quite expensive. If it does not have the 'click of death' as described, then it's probably an enclosure failure, which takes us to step 4.
4) Replace the enclosure of the drive. Proceed with extreme caution. Errors made in replacing the enclosure may cause data loss! You can buy external drive enclosures at your local computer store. Take your old drive with you so that they can determine which enclosure will suit you best. Online retailers also sell them, usually at a lower cost. You'll need to remove the hard drive from the old enclosure and install it in the new one. There are various tutorials online. A Google search for "external drive replace enclosure" (without the quotes) should suffice to get you going in the right direction.
If you aren't comfortable with the troubleshooting, or with the enclosure replacement, then by all means take it to a pro. They should be able to determine the cause and the solution rather quickly.