In Windows, I want to enter a key command and have whatever window is currently beneath the cursor be made the active window. Is there a piece of software that makes this possible?
The style of mouse-window interaction you are looking for is called focus follows mouse pointer, and on Windows it's also called x-mouse or window tracking.
Windows XP: I'd recommend Microsoft's Tweak UI, and the X-Mouse settings there.
Windows 7: There are settings to enable it in the Ease of Access Center (Win+U), but to tweak the delay I recommend my small open source utility X-Mouse Controls.
X-Mouse Controls: Windows utility to enable or disable active window tracking, raising and also the delay in milliseconds. This is known as x-mouse behavior or focus follows mouse.
Window raising is discouraged. A delay of 100-250 ms should work well.