Missing External Hard Drive

I would try rebooting your Mac,trying another USB cord. Did you give it a minute to recognize the drive?

 

Sounds like it could be a bad drive. Do other usb devices work on the same MBP port? Have you tried the drive with another computer recently?

On a side note, if you're expanding to an external drive, make sure your backup device has the capacity to accomodate the additional storage.

 

"When I connect the external hard drive to the mac, nothing happens,
It doesn't show in Finder, on the Desktop, or, more worryingly, in Disk Utility."

This is a USB drive, is that correct?

When you connect the drive to the MacBook, put your ear to it.

Do you hear the drive's platter spin up inside?
Do you hear anything at all?

Assuming you do hear the drive spin up (or possibly spin up for a few moments and then "spin back down") -- I'm going to guess that the drive requires more "USB bus power" than your MacBook can supply. This is a common problem with MacBooks and USB bus-powered portable drives.

Sometimes, the drive may act like it's spinning up, but never fully "comes online", thus -- no drive icon on the Finder desktop, no indication that it's connected when you open Disk Utility.

If you connect the drive and open System Profiler, does it show up on the list of USB devices?

Does the drive enclosure have a port for connecting some kind of external power supply (usually a "power brick")?

If you have a power supply, have you tried using it with the drive?

There is also a "Y-cable" that can be bought, that will sometimes help in such instances by "doubling" the USB port connections (on the Mac end), and thus supplying a bit more USB power.

A last resort (actually, not a bad one, IF the drive is SATA) is to take the drive out of its enclosure, and instead use a gadget like this:
http://www.amazon.com/s/ref=nb_sb_noss?url=search-alias=aps&field-keywords=usb+sata+dock&x=0&y=0
(many items shown, they all work the same, just pick one you like that's cheap)
A USB/SATA dock will come in handy for other uses, as well.