How do I install Linux on a hard drive from within Windows? I'm
wondering if there is an application that runs entirely under Windows
that will allow me to install a bootable Linux (or other OS) partition
on a hard drive attached to the system, such as one connected
temporarily through a USB port.

Technically yes

One would use a Windows based disk imaging system to create/write the partition
It is not really an 'installation'

Check out Mandrake. They have a good Windows-based installation setup.
Also, if you want to control the partition modifications, check out
Partition Magic. Runs in Windows, can resize your partition easily
without loosing any data for future installation of Linux.

Yes it is entirely possible, infact their are linux distribution which
work entirely within windows, besides, one can also get a special
version of the linux kernel, designed to work 'on' windows.

Linux Application can also be entirely compiled from with windows

Some related links:
The kernel version I was talking off
http://www.colinux.org/

some screenshots, of colinux in action:
http://www.colinux.org/?section=screenshots

The Linux based Operation System (Distribution) designed to work
completely under windows
http://www.topologilinux.com

and the customary screenshots:
http://www.topologilinux.com/index.php?menu=6

You can use Visual PC, a product by MicroSoft, and Vmware.

I would use VMWare (it's not free) but it is very good. MS's virtual
pc is crap when I used it.

I believe that the question asker is wishing just an INSTALLATION
within windows, not an operating kernel within windows.

For the record, TopologiLinux, the distro referred to by irfanhab, I
have never had any success with, mainly due to its use of a custom
network driver to begin the X Session.