An app is a program that you manually open, it takes up the entire screen (or half the screen in some Samsungs) and does its thing.
A widget sits on the screen, the size of 1, 2, 3, etc., icons and does its thing. Like a clock that sits on the screen constantly giving you the time, or a weather widget that constantly gives you the temperature. (Windows 7 had something similar you could add to the desktop. Maybe Vista did too, but I try to forget as much about Vista as I can.)
They're both programs, they both do the same things in the same way (a clock app and a clock widget get the time from the same place), the difference is basically how you use them and how they display.