Well, a quick and dirty way to say is that the Widget is basically a more interactive type icon on your Phone's Home screens. Normally with apps, you 'delve' into the app, it basically has you opening the app and taking the entire screen. Widgets, on the other hand, can interact with the app it is associated with or be like an App showing you information.
A better way to explain it is looking at the widgets themselves in comparison to the app:
Music - The app calls up the window to show options on playing music, literally like an MP3 player where you do browsing, playlists and the sort.
Music Widget - It's a bar Icon you place on the home screen, but instead of opening up the Music App, your options are play or next track, showing the song based on your last preference. You can tell it to pause on the track or go next track, but if you want to do more, simply tapping on the widget track part, it delves into the Music app itself for more control.
Each widget will have different ways to handle things, like I use the battery widget to give me a percent and visual reading of the battery versus going off the one on the notification bar. If I tap on it, it allows me to toggle or jump to certain settings. Another Battery one, if you tap a certain location it, it will pull up a different menu to look at history and stuff.
The key thing with Widgets, is that they do a little more than just be an icon or shortcut to an application.
