If your external hard drive enclosure is combo USB / eSATA, I found this info in the directions for mine:
"Connect the eSATA cable BEFORE connecting the USB cable to the enclosure, or the enclosure will default to USB.
* You need to connect the USB cable to draw power from either your PC or Laptop."
The eSATA connection is for data only. The power for the enclosure's circuits / the hard drive inside it is provided by the USB cable.
The USB port you plug into MUST be one that the external drive can work properly with when plugged into, and the connection MUST be able to supply the full USB spec 500ma. See response 5.
Out of curiousity, I tried connecting my combo USB / eSATA external drive via a eSATA connection.
With the AC power to the case ON, the computer off, I connected the eSATA cable between the computer and the enclosure - no led lit up on the external enclosure - then I connected the USB cable between the computer and the enclosure - an led came on, on the external enclosure.
(Note that for my external enclosure, the led will bink if the USB port cannot supply 500ma, or if there is some other problem with being connected to that USB port.)
Then I booted the computer (into Vista), eventually a message popped up saying a device had been detected - the name and model of the external enclosure's circuitry. Then a message said installing software for it or similar. I looked in Computer and the three partitions on the eSATA connected drive were there. I looked in Device Manager and under Disk drives the hard drive model of the drive inside the external enclosure was listed and had the icon for a removable (external) drive. I looked in Disk Management - the hard drive model inside the external enclosure was there, and the three partitions on it were shown.
There was a message window on the desktop screen that said I need to restart Windows because of changes made or similar (the external enclosure's circuits being detected and software - drivers - having been installed for it) so I did that, and while booting I went into the bios Setup - the hard drive model inside the external eSATA connected enclosure is listed there.