Power from USB port? Generally, no.
Virtually all desktop-type HDD's, and surely most older ones, need more power than is available from a USB2 port. In almost all cases, enclosures and complete external drives that do draw all their power from a USB2 port are the smaller units that are normally found in laptops. Even some of those require two USB ports used simultaneously.
IF you get an enclosure that uses an eSATA or an IEEE1394 (Firewire 400) connection, neither of these has any power available for an external device, so it certainly will come with its own power supply. In fact, many external enclosures come with two (sometimes three) interface options, but you get to use only one at a time. Mine has eSATA and USB2, for example.
The new USB3 ports can provide more power than USB2, but doing that would limit your possible use to only those machines that have this new port. I'd say you are better to choose an enclosure that has its own power supply and does not use any from a USB port. Then you can use it on anything.